This blog has not only required extensive amounts of time, but has opened many outlets for me to expand my knowledge in the study of physics. Both this acquired knowledge and blog will serve as a resource I can easily access during my four years at WPI (Worcester Polytechnic Institute). Thank you, Mr. Connors, for providing me with this unique opportunity. College...here I come!
Philippe Kelley - Northbridge High School class of 2012
Friday, May 18, 2012
Lesson 46 The Engine of Nature
The invention of the steam engine led to the expansion of western culture due to both transportation and economic benefits.
James Watt used a condenser to cool steam outside of the engine allowed for further developments and progress.
A cylinder with a moveable end, pushed by pressurized steam, opening a valve to the boiler emits high pressure steam that pushes the piston out doing work on the fly wheel, which pushes the cylinder back in again for the next cycle.
The water wheel led to the steam engine, where water or steam can only flow from a higher plane to a lower plane. Sadi Carnot theorized the steam engine must be comprised of mechinisms that allow for the flow of heat to progress form high coloric to low coloric.
No machine or combination of machines can ever have the effect of making more heat run up to high temperature than down to low temperature. This is known as the second law of thermodynamics.
The most effective engine is one that can transform low temperatures to high temperatures.
Isothermal - as heat is applied, gas expands and work is done. The cylinder is heated prior to the application of more heat. This will allow for a reversable action as heat may flow in both directions.
Adiabatic - a conversion that occurs without input or release of heat within a system.
In a Carnot engine, the engine follows a process that is repeated:
Isothermal - the cylinder is heated prior to the application of more heat which allows for the expansion of the gas.
Adiabatic - no heat is applied as the gas continues to expand as the heat of the cylinder continues to heat the gas.
Isothermal - the cylinder is cooled prior to the application of low temperature which allows for the gas to contract.
Adiabatic - the cylinder is not cooled as the cooled cylinder causes the gas and piston to contract nearly completely.
Gases need space to expand and contract, and require a container for heat to be released.
Carnot engine:
e (efficiency) = W/Qi
The most efficient engine that nature allows follows the principle Qo/Qi = To/Ti. These discoveries led to the future concept of entropy.
James Watt used a condenser to cool steam outside of the engine allowed for further developments and progress.
A cylinder with a moveable end, pushed by pressurized steam, opening a valve to the boiler emits high pressure steam that pushes the piston out doing work on the fly wheel, which pushes the cylinder back in again for the next cycle.
The water wheel led to the steam engine, where water or steam can only flow from a higher plane to a lower plane. Sadi Carnot theorized the steam engine must be comprised of mechinisms that allow for the flow of heat to progress form high coloric to low coloric.
No machine or combination of machines can ever have the effect of making more heat run up to high temperature than down to low temperature. This is known as the second law of thermodynamics.
The most effective engine is one that can transform low temperatures to high temperatures.
Isothermal - as heat is applied, gas expands and work is done. The cylinder is heated prior to the application of more heat. This will allow for a reversable action as heat may flow in both directions.
Adiabatic - a conversion that occurs without input or release of heat within a system.
In a Carnot engine, the engine follows a process that is repeated:
Isothermal - the cylinder is heated prior to the application of more heat which allows for the expansion of the gas.
Adiabatic - no heat is applied as the gas continues to expand as the heat of the cylinder continues to heat the gas.
Isothermal - the cylinder is cooled prior to the application of low temperature which allows for the gas to contract.
Adiabatic - the cylinder is not cooled as the cooled cylinder causes the gas and piston to contract nearly completely.
Gases need space to expand and contract, and require a container for heat to be released.
Carnot engine:
e (efficiency) = W/Qi
The most efficient engine that nature allows follows the principle Qo/Qi = To/Ti. These discoveries led to the future concept of entropy.
Lesson 45 Temperature and Gas Laws
Heat is the random motion of atoms and particles.
Temperature can only be measured in terms of its effects.
0 degrees celsius - water freezes
100 degrees celsius - water boils
32 degrees fahrenheit - water freezes
212 degrees fahrenheit - water boils
Temperature also affects pressure. Pressure is the force per unit area exerted.
Pressure = force/area
A molecule changes momentum when it exerts force on the wall of a balloon. The opposite forces results in the expansions of a balloon.
Heating a gas causes an increase in kinetic energy of the molecules, resulting in more pressure. The pressure of a gas is proportional to the number of molecules, inversely proportional to the volume, and proportional to the average kinetic energy of the molecule. The constant of proportionality is 2/3.
p = 2/3 (N/V) k
pv = constant if temperature is constant
p1v1 = p2v2 Boyle's Law
_
constant = 2/3 Nk
pv is proportional to the kinetic energy of all molecules of the gas. This kinetic energy is a form of heat. Heating a gas causes pressure to rise and volume to rise.
All gasses expand by the same amount with a given rise in temperature. At a given pressure the volume of gas changes by the same fraction for each degrees in temperature.
v1/t1 = v2/t2 Charles's Law
There exists a temperature so low that a gas would fill no volume. This temperature is known as absolute zero, where a gas has no heat.
Absolute zero is -273 degrees celsius, -459 degrees fahrenheit, and 0 degrees kelvin.
pv = knt determines the size of one kelvin where k = 1.38 x 10^-23 J/K
Kinetic Theory of Gasses
The kinetic energy of a gas, or the collective effects of molecular collisions, is what gives a gas pressure and volume.
Temperature and heat connection:
_
knt = (2/3) Nk
pv = knt
_
pv = (2/3) nk
_
kt = (2/3) k - absolute temperature
Heat and temperature are related by kinetic energy of the molecules of gas.
Heat in a gas is the average kinetic energy of molecules.
Absolute temperature is (2/3) the mean average kinetic energy of one molecule of gas
pv = Nkt - all temperature and pressure are related by the ideal gas law, which leads to the kinetic theory.
Temperature can only be measured in terms of its effects.
0 degrees celsius - water freezes
100 degrees celsius - water boils
32 degrees fahrenheit - water freezes
212 degrees fahrenheit - water boils
Temperature also affects pressure. Pressure is the force per unit area exerted.
Pressure = force/area
A molecule changes momentum when it exerts force on the wall of a balloon. The opposite forces results in the expansions of a balloon.
Heating a gas causes an increase in kinetic energy of the molecules, resulting in more pressure. The pressure of a gas is proportional to the number of molecules, inversely proportional to the volume, and proportional to the average kinetic energy of the molecule. The constant of proportionality is 2/3.
p = 2/3 (N/V) k
pv = constant if temperature is constant
p1v1 = p2v2 Boyle's Law
_
constant = 2/3 Nk
pv is proportional to the kinetic energy of all molecules of the gas. This kinetic energy is a form of heat. Heating a gas causes pressure to rise and volume to rise.
All gasses expand by the same amount with a given rise in temperature. At a given pressure the volume of gas changes by the same fraction for each degrees in temperature.
v1/t1 = v2/t2 Charles's Law
There exists a temperature so low that a gas would fill no volume. This temperature is known as absolute zero, where a gas has no heat.
Absolute zero is -273 degrees celsius, -459 degrees fahrenheit, and 0 degrees kelvin.
pv = knt determines the size of one kelvin where k = 1.38 x 10^-23 J/K
Kinetic Theory of Gasses
The kinetic energy of a gas, or the collective effects of molecular collisions, is what gives a gas pressure and volume.
Temperature and heat connection:
_
knt = (2/3) Nk
pv = knt
_
pv = (2/3) nk
_
kt = (2/3) k - absolute temperature
Heat and temperature are related by kinetic energy of the molecules of gas.
Heat in a gas is the average kinetic energy of molecules.
Absolute temperature is (2/3) the mean average kinetic energy of one molecule of gas
pv = Nkt - all temperature and pressure are related by the ideal gas law, which leads to the kinetic theory.
Lesson 44 Energy, Momentum, and Mass
Einstein developed the Theory of Relativity, to which momentum also applies.
A ball will not come to rest until something stops it.
Po = P1 + P2
Po^2 = P1^2 + P2^2
If one ball is at relative rest:
Po = P2
Po^2 = P2^2
When objects collide the change in force is equal and opposite, so is momentum.
p = mv
F = d/dt(p)
F1 = -F2
dp1/dt = -dp2/dt
d/dt (p1 + p2) = 0 the total momentum of both balls remains constant.
Momentum remains contant in Einstein's Theory of Relativity.
In outer space, the momentum of two colliding objects released while in motion is equal and opposite as on earth.
M delta(U) = delta(P)
However, according to the theory of relativity, a collision of two balls occurs and results in equal momentum where less velocity is accounted for with greater mass. This increase in mass is influenced by speed.
F = ma
Mass is an object's resistance to change in velocity.
According to the Theory of Relativity, the mass of a moving object depends on how fast it is moving. The mass m is equal to mo (rest mass) x gamma.
Mass, like time and distance, depends on one's point of view. An object at rest resists less than object in motion. Mass depends on speed, but no mass can reach the speed of light. Inside any accelerator electromagnetic fields exert force on a tiny ion. This force increases the ion's momentum mo(v), so the speed increases and so does the mass. As objects become more massive, it becomes more difficult to make them accelerate. The ion's momentum and energy continue to increase but the speed never reaches the speed of light.
Increasing the kinetic energy of a bod increases its mass.
x1
W = integral (F) dx
xo
Kinetic energy or force changes momentum.
K = integral dx/dt dp p = mv
= integral v dp dp = mo (1/(1-v^2/c^2)) dv
K = integral v mo (1/(1-v^2/c^2))^(3/2) dv
K = mo (c^2/(1-v^2/c^2)^(1/2)) - moc^2
K = (m - mo) c^2
Eo = moc^2 - rest mass energy
E = mc^2 - total energy of the body
A loss in energy accounts for a loss in mass.
A ball will not come to rest until something stops it.
Po = P1 + P2
Po^2 = P1^2 + P2^2
If one ball is at relative rest:
Po = P2
Po^2 = P2^2
When objects collide the change in force is equal and opposite, so is momentum.
p = mv
F = d/dt(p)
F1 = -F2
dp1/dt = -dp2/dt
d/dt (p1 + p2) = 0 the total momentum of both balls remains constant.
Momentum remains contant in Einstein's Theory of Relativity.
In outer space, the momentum of two colliding objects released while in motion is equal and opposite as on earth.
M delta(U) = delta(P)
However, according to the theory of relativity, a collision of two balls occurs and results in equal momentum where less velocity is accounted for with greater mass. This increase in mass is influenced by speed.
F = ma
Mass is an object's resistance to change in velocity.
According to the Theory of Relativity, the mass of a moving object depends on how fast it is moving. The mass m is equal to mo (rest mass) x gamma.
Mass, like time and distance, depends on one's point of view. An object at rest resists less than object in motion. Mass depends on speed, but no mass can reach the speed of light. Inside any accelerator electromagnetic fields exert force on a tiny ion. This force increases the ion's momentum mo(v), so the speed increases and so does the mass. As objects become more massive, it becomes more difficult to make them accelerate. The ion's momentum and energy continue to increase but the speed never reaches the speed of light.
Increasing the kinetic energy of a bod increases its mass.
x1
W = integral (F) dx
xo
Kinetic energy or force changes momentum.
K = integral dx/dt dp p = mv
= integral v dp dp = mo (1/(1-v^2/c^2)) dv
K = integral v mo (1/(1-v^2/c^2))^(3/2) dv
K = mo (c^2/(1-v^2/c^2)^(1/2)) - moc^2
K = (m - mo) c^2
Eo = moc^2 - rest mass energy
E = mc^2 - total energy of the body
A loss in energy accounts for a loss in mass.
Lesson 43 Velocity and Time
If one uses a loop of wire and a galvanometer a light will move when an electric current passes through the wire. This electric current can be created by moving and removing a bar magnet from the loop of wire. The force that sets electric charges in a wire in to motion are described by the equation below:
F = qE + qv x B
Einstein desired to further explain why this equation remains true and developed the Theory of Relativity and found that time is relative:
delta(t) = gamma delta (t)o
gamma = 1/sqrt(t - v^2/c^2)
According to the Theory of Relativity, timed events occur more slowly by a factor of gamma, depending on the position of the observer. The Theory of Relativity is a theory of motion based upon different observers where there is no absolute motion and no absolute rest.
Nothing travels faster than the speed of light. If an observer is detecting the speed of a baseball thrown from a moving platform the speed or velocity cn be determined by the slope of the observer and baseball in the space-time continuum where both space and time are independent axes.
v' = delta(x)'/delta(t)'
Speeds computed in the spacetime frame are always less than one, or the speed of light.
Ux = (U'x + v)/(1+ VU'x/c^2)
Even along the direction of relative motion, components of velocity never add to a speed greater than the speed of light.
Uy = U'y/(gamma(1 + VU'x/c^2))
Uz = U'z/(gamma(1 + VU'x/c^2))
The veocity transformation does affect components perpendicular to the motion because although distances remain the same, time changes from one frame to the next.
There is a test of relativity in the phenomenon decay of cosmic ray muons.
Because time is relative, if one travels close to the speed of light to an area of space ten lightyears away, due to Lorentz contractions, the individual travelling only ages several months. An individual on earth ages 20 years.
In a completely empty universe time and space have no relevance. In a universe with many bodies, no object is at complete rest, but remains in motion until an object acts upon it, following the law of inertia. Time and space are relevant in this universe. Rest can only be observed in frames of time and space.
F = qE + qv x B
Einstein desired to further explain why this equation remains true and developed the Theory of Relativity and found that time is relative:
delta(t) = gamma delta (t)o
gamma = 1/sqrt(t - v^2/c^2)
According to the Theory of Relativity, timed events occur more slowly by a factor of gamma, depending on the position of the observer. The Theory of Relativity is a theory of motion based upon different observers where there is no absolute motion and no absolute rest.
Nothing travels faster than the speed of light. If an observer is detecting the speed of a baseball thrown from a moving platform the speed or velocity cn be determined by the slope of the observer and baseball in the space-time continuum where both space and time are independent axes.
v' = delta(x)'/delta(t)'
Speeds computed in the spacetime frame are always less than one, or the speed of light.
Ux = (U'x + v)/(1+ VU'x/c^2)
Even along the direction of relative motion, components of velocity never add to a speed greater than the speed of light.
Uy = U'y/(gamma(1 + VU'x/c^2))
Uz = U'z/(gamma(1 + VU'x/c^2))
The veocity transformation does affect components perpendicular to the motion because although distances remain the same, time changes from one frame to the next.
There is a test of relativity in the phenomenon decay of cosmic ray muons.
Because time is relative, if one travels close to the speed of light to an area of space ten lightyears away, due to Lorentz contractions, the individual travelling only ages several months. An individual on earth ages 20 years.
In a completely empty universe time and space have no relevance. In a universe with many bodies, no object is at complete rest, but remains in motion until an object acts upon it, following the law of inertia. Time and space are relevant in this universe. Rest can only be observed in frames of time and space.
Lesson 42 The Lorentz Transformation
The Michelson-Morley experiment set out to determine the existence of the luminiferous ether, through which the speed of the earth was belived to be determined. Michelson and Morley attempted to determine the existence of the ether through its effects on the speed of light.
The reason both light beams reach the interferometer at the same time is due to the contraction of one of the device's arms. Lorentz supported this idea with the theory of the properties of electrons where electrons contract in the direction of motion. This led to the idea of relativity where the speed of an object is determined through the speed of the observer.
Time and distance are affected by motion.
The speed of light is the same for all observers. However, if light is in motion between two mirrors, the observer will see the light taking less time to complete one cycle. This is due to the increased diagonal distance.
The relativity of time is derived from the right triangle formed by the distances traveled.
The Pythagorean theorem shows that the path of the moving light is longer than the distance between the two mirrors.
delta(t) = 1/sqrt(1-(v^2/c^2))delta(t)'
delta(t) = gamma delta(t)'
Two observers can agree on the occurence of an event with respect to time using the Lorentz transformations
Galileo - x' = x - vt
Lorentz - x' = gamma( x - vt) the equation along the direction of motion.
y' = y the equation describing the perpendicular motion that is the same in both frames of observation.
z' = z
t' = gamma (t - vx/c^2)
The Lorentz transformations link time and space together.
Einstein developed two postulates:
1st - the laws of physics are the same for all inertial frames
2nd - the speed of light is the same for all observers.
The reason both light beams reach the interferometer at the same time is due to the contraction of one of the device's arms. Lorentz supported this idea with the theory of the properties of electrons where electrons contract in the direction of motion. This led to the idea of relativity where the speed of an object is determined through the speed of the observer.
Time and distance are affected by motion.
The speed of light is the same for all observers. However, if light is in motion between two mirrors, the observer will see the light taking less time to complete one cycle. This is due to the increased diagonal distance.
The relativity of time is derived from the right triangle formed by the distances traveled.
The Pythagorean theorem shows that the path of the moving light is longer than the distance between the two mirrors.
delta(t) = 1/sqrt(1-(v^2/c^2))delta(t)'
delta(t) = gamma delta(t)'
Two observers can agree on the occurence of an event with respect to time using the Lorentz transformations
Galileo - x' = x - vt
Lorentz - x' = gamma( x - vt) the equation along the direction of motion.
y' = y the equation describing the perpendicular motion that is the same in both frames of observation.
z' = z
t' = gamma (t - vx/c^2)
The Lorentz transformations link time and space together.
Einstein developed two postulates:
1st - the laws of physics are the same for all inertial frames
2nd - the speed of light is the same for all observers.
Lesson 41 The Michelson-Morley Experiment
The Michelson-Morley Experiment was designed to detect the motion of the earth through the luminiferous ether.
The ether was created to explain how light travels from the sun to the earth, as light waves propogate from the sun through this medium.
The experiment proved there was no ether, however, Einstein developed the Theory of Relativity from its failure, which explained what the experiment attempted to do.
Due to the Theory of Relativity and the failure of the Michelson-Morley experiment, space was now recognized as a vacuum.
If the ether was viscous, planets would lose energy and spiral inward towards the sun. Therefore, physicists believed the ether was a perfectly mobile fluid with no viscosity. It is virtually incompressible, transparent, and fills all of space allowing planets to obey Newton's laws with ease.
Michelson developed the interferometer that splits a light in two beams. As light travels through a partially transparent and opaque mirror, light is sent in two beams in perpendicular directions. The light is then sent back by two opaque mirrors and the light combines into one beam. If the beams of light ravel at the same speed, the interferometer produces a read-out of a spiral with a single dot in the center.
The speed of light c = 3 x 10^8 m/s
The speed of the earth through space v = 3 x 10^4 m/s
Difference in completion time (v/c)^2 = 10^-8
Lorentz developed the Lorentz transformations (mathematical equations) to discuss the phenomenon of the retraction of a mirror in the interferometer.
Henri Poincare developed the priciple of relativity, that absolute motion will never be observed in a laboratory.
The ether was created to explain how light travels from the sun to the earth, as light waves propogate from the sun through this medium.
The experiment proved there was no ether, however, Einstein developed the Theory of Relativity from its failure, which explained what the experiment attempted to do.
Due to the Theory of Relativity and the failure of the Michelson-Morley experiment, space was now recognized as a vacuum.
If the ether was viscous, planets would lose energy and spiral inward towards the sun. Therefore, physicists believed the ether was a perfectly mobile fluid with no viscosity. It is virtually incompressible, transparent, and fills all of space allowing planets to obey Newton's laws with ease.
Michelson developed the interferometer that splits a light in two beams. As light travels through a partially transparent and opaque mirror, light is sent in two beams in perpendicular directions. The light is then sent back by two opaque mirrors and the light combines into one beam. If the beams of light ravel at the same speed, the interferometer produces a read-out of a spiral with a single dot in the center.
The speed of the earth through space v = 3 x 10^4 m/s
Difference in completion time (v/c)^2 = 10^-8
Lorentz developed the Lorentz transformations (mathematical equations) to discuss the phenomenon of the retraction of a mirror in the interferometer.
Henri Poincare developed the priciple of relativity, that absolute motion will never be observed in a laboratory.
Lesson 40 Optics
Light is a wavelength disturbance of the electromagnetic field that propogates at a definite speed. Light has properties similar to those of all waves, waves can propogate outward from a single point of disturbance. Waves from carefully coordinated arrays of point sources can add up to form wave fronts called plane waves. Plane waves can be made to spread out again because waves bend around corners. When wave fronts encounter one another, they can produce stronger and weaker waves.
Galileo not only developed one of the first telescopes, but used it most practically. Galileo also developed the compound microscope.
Lenses of glasses, telescopes, and microscopes are based off the principle of refraction. Refraction occurs when light is bent as it travels through a medium. Manufacturers of glasses, telescopes, and microscopes develop convex lenses that refract light to a single point.
A glass prism not only bends or refracts a beam of light, but separates white light into the colors of the visible light spectrum. This is known as dispersion.
Newton theorized that reflection and refraction can be explained by the gravitational attraction between matter, however, light was composed of waves rather than particles.
Electromagnetic waves are always transverse and travel at the speed of light, and can have different frequencies and wavelengths that create the visible light spectrum. The visible light spectrum is composed of waves with lengths between 400 and 700 nanometers.
When electric charges with lines of force are set into vibration oscillating electric charges create waves that propogate along the lines of force at the speed of light. These waves become closer and closer in proximity and become wave fronts, as they become flatter farther away from the source and develop into plane waves. An electric charge is the source of outward radiation in the electric field. Thus light travels in waves.
Light has the property of waves interference and can either be constructive or destructive. If waves are in step they reinforce each other and are contructive. If waves are out of step they cancel each other and are destructive.
When light waves encounter electric charges the oscillating electric field makes the charge oscillate which creates a new outward travelling wave.
In metals, where electrons move freely, light can be reflected where the angle of incidence is equal to the angle of reflection.
Glasses function because the speed of light is slower in the glass than in the air. This causes light to refract and concentrate towards the center of the lens on the retina.
Galileo not only developed one of the first telescopes, but used it most practically. Galileo also developed the compound microscope.
Lenses of glasses, telescopes, and microscopes are based off the principle of refraction. Refraction occurs when light is bent as it travels through a medium. Manufacturers of glasses, telescopes, and microscopes develop convex lenses that refract light to a single point.
A glass prism not only bends or refracts a beam of light, but separates white light into the colors of the visible light spectrum. This is known as dispersion.
Newton theorized that reflection and refraction can be explained by the gravitational attraction between matter, however, light was composed of waves rather than particles.
Electromagnetic waves are always transverse and travel at the speed of light, and can have different frequencies and wavelengths that create the visible light spectrum. The visible light spectrum is composed of waves with lengths between 400 and 700 nanometers.
When electric charges with lines of force are set into vibration oscillating electric charges create waves that propogate along the lines of force at the speed of light. These waves become closer and closer in proximity and become wave fronts, as they become flatter farther away from the source and develop into plane waves. An electric charge is the source of outward radiation in the electric field. Thus light travels in waves.
Light has the property of waves interference and can either be constructive or destructive. If waves are in step they reinforce each other and are contructive. If waves are out of step they cancel each other and are destructive.
When light waves encounter electric charges the oscillating electric field makes the charge oscillate which creates a new outward travelling wave.
In metals, where electrons move freely, light can be reflected where the angle of incidence is equal to the angle of reflection.
Glasses function because the speed of light is slower in the glass than in the air. This causes light to refract and concentrate towards the center of the lens on the retina.
Friday, April 6, 2012
Lesson 39 Maxwells's Equations
Maxwell built upon Faraday's discoveries concerning lines of force of both electric charges and magnetic poles through the application of mathematics.
Every wave has a particular apeed:
Water: v = sqrt(qy/2pi)
Sound: v = sqrt(p/dr)
Linked oscillators: v = sqrt(k/m)
Maxwell attempted to determine the speed of waves in Faraday's lines of force. These forces were similar to the following equations:
F(g) = -G m1m2/r^2 (r^)
F(e) = ke q1q2/r^2 (r^)
F(m) = km p1p2/r^2 (r^)
Specific constants:
G = 6.7 x 10^-11 Nm^2/kg^2
ke = 9 x 10^9 Nm^2/ c^2
km = 1 x 10^-7 Ns^2/c^2
Since these equations are not independent, ke and km are related:
ke/km = 3 x 10^8 m/s^2 or the speed of light where both magnetic and electric waves propogate at the speed of light.
The medium through which these waves propagate is the electromagnetic field, and they obey these equations:
0integral(integral( B dA)) = 0
E(o)d/dtintegral(integral( E dA)) = I
0integral( B dr) = u(o) (I + E(o)d/dt integral(integral(E dA)))
0integral( E dr) = 0
With every electric wave, there is also a magnetic wave.
Every wave has a particular apeed:
Water: v = sqrt(qy/2pi)
Sound: v = sqrt(p/dr)
Linked oscillators: v = sqrt(k/m)
Maxwell attempted to determine the speed of waves in Faraday's lines of force. These forces were similar to the following equations:
F(g) = -G m1m2/r^2 (r^)
F(e) = ke q1q2/r^2 (r^)
F(m) = km p1p2/r^2 (r^)
Specific constants:
G = 6.7 x 10^-11 Nm^2/kg^2
ke = 9 x 10^9 Nm^2/ c^2
km = 1 x 10^-7 Ns^2/c^2
Since these equations are not independent, ke and km are related:
ke/km = 3 x 10^8 m/s^2 or the speed of light where both magnetic and electric waves propogate at the speed of light.
The medium through which these waves propagate is the electromagnetic field, and they obey these equations:
0integral(integral( B dA)) = 0
E(o)d/dtintegral(integral( E dA)) = I
0integral( B dr) = u(o) (I + E(o)d/dt integral(integral(E dA)))
0integral( E dr) = 0
With every electric wave, there is also a magnetic wave.
Lesson 38 Alternating Current
Alternating current (AC) is an oscillating current. It is created by a voltage that rises and falls.
Direct current (DC) is a steady flow of electricity where, idealy, its voltage is constant and equal to the current, that is also constant, multiplied by effective resistance of the circuit.
Consider a circuit that consists of an alternating voltage source, a capacitor, and an inductor. According to the mathematical rules of Gustav Kircchoff, the rises in voltage at the source is never greater than E(o) which is equivalent to the sum of voltage drops around the circuit.
E(o)sinwt = LdI/dt + q/c
The result is a differential equation that can be written in terms of charge q on the capacitor.
= Ld^2/dt^2 + q/c
E(o)sinwt = m(Ld^2/dt^2) + kx this same equation describes the displacement x of a harmonic oscillator.
Both equations lead to resonance, which, in a circuit can cause the flow of a great amount of charge.
Radio and television signals are transfered through electric resonance.
A capacitor opposes change in positive or negative charge, while an inductor opposes change in current in the same manner the intertial mass on a spring opposes change in velocity.
In an AC, if the frequency is low enough the charging and discharging of the capacitor can keep up with the oscillating applied voltage. At higher frequency, the capacitor cannot charge and discharge fast enough. Therefore, no voltage difference develops across it and nearly all voltage is across the resistor.
If an inductor is in the circuit at low frequency there is time for voltage to build within it and not across. In high frequency the voltage cannot change fast enough so most of the voltage is across the inductor.
When all elements are in the same circuit at low frequency, most voltage charges and discharges the capacitor. At higher frequency, most of the voltage is used changing the current in the inductor. If the frequency is at the resonant frequency, quite a large current flows.
Power = current x voltage
P = IV
Heating = P^2R/V^2
Therefore, the higher the voltage the less power is lst in transit over long distances along electric wires.
If AC passes through a loop of coil, a constantly changing magnetic flux is produced. If a set of coils of wires are wrapped around an iron loop, the flux is completely contained. This flux produces voltage.
Voltage can be increased or decreased safely with this method due to the fact that voltage is proportional to the number of coils of wire around the iron loop.
This is the reason AC won over DC in the modern world.
Direct current (DC) is a steady flow of electricity where, idealy, its voltage is constant and equal to the current, that is also constant, multiplied by effective resistance of the circuit.
Consider a circuit that consists of an alternating voltage source, a capacitor, and an inductor. According to the mathematical rules of Gustav Kircchoff, the rises in voltage at the source is never greater than E(o) which is equivalent to the sum of voltage drops around the circuit.
E(o)sinwt = LdI/dt + q/c
The result is a differential equation that can be written in terms of charge q on the capacitor.
= Ld^2/dt^2 + q/c
E(o)sinwt = m(Ld^2/dt^2) + kx this same equation describes the displacement x of a harmonic oscillator.
Both equations lead to resonance, which, in a circuit can cause the flow of a great amount of charge.
Radio and television signals are transfered through electric resonance.
A capacitor opposes change in positive or negative charge, while an inductor opposes change in current in the same manner the intertial mass on a spring opposes change in velocity.
In an AC, if the frequency is low enough the charging and discharging of the capacitor can keep up with the oscillating applied voltage. At higher frequency, the capacitor cannot charge and discharge fast enough. Therefore, no voltage difference develops across it and nearly all voltage is across the resistor.
If an inductor is in the circuit at low frequency there is time for voltage to build within it and not across. In high frequency the voltage cannot change fast enough so most of the voltage is across the inductor.
When all elements are in the same circuit at low frequency, most voltage charges and discharges the capacitor. At higher frequency, most of the voltage is used changing the current in the inductor. If the frequency is at the resonant frequency, quite a large current flows.
Power = current x voltage
P = IV
Heating = P^2R/V^2
Therefore, the higher the voltage the less power is lst in transit over long distances along electric wires.
If AC passes through a loop of coil, a constantly changing magnetic flux is produced. If a set of coils of wires are wrapped around an iron loop, the flux is completely contained. This flux produces voltage.
Voltage can be increased or decreased safely with this method due to the fact that voltage is proportional to the number of coils of wire around the iron loop.
This is the reason AC won over DC in the modern world.
Lesson 37 Electromagnetic Induction
Electromagnetic induction is the basis for all electrical advancement and electrical discoveries.
Michael Faraday developed the world's first electric motor, where a charged wire follows the circular magnetic field created by the electric current.
Electromagnetic induction is how a magnetic field drives an electric current around a circuit.
Magnetic fields apply forces to electric charges, but only if the charges are in motion.
F=qV x B
The direction of the force is perpendicular to the velocity and the field, and depends on the sign of the charge (negative in this case). Moving a wire charged with electrons through a magnetic field causes the current to flow.
The field of a bar magnet can be perpendicular to every point of a circular loop of wire. Moving the loop drives a current around the loop. Moving the loop up drives the current one way, and moving the loop down drives the current another way. This also applies if the magnet is moved.
Any method of changing magnetic field makes current flow. This causes a change in magnetic flux which changes the current flow as if by voltage. E = d (IO) or Faraday's law. of electro magnetic induction.
Continually rotating a loop of wire in a magnetic field changes the current in the loop. The resulting voltage and current are sinusoidal, going first one way and then the other.
Current flows through a coil of wire when the magnetic flux through it is changing, the current flows one way if the flux decreases and the other way when the flux increases. The current induced in the wire creates a flux of its own, whose direction depends on the dircetion of the current. The flux created by the induced current always opposes a change in the external flux. This is known as Lenz'z law.
An electric current in any circuit creates a magnetic field which whenever it changes induces a current in the same circuit that opposes changes. This is called self-induction.
The induction of a circuit element:
E = -L dI/dt L= (dIO)/dI
Both resstors and capacitor exhibit self-induction. Electric fields will only circulate if magnetic flux is changed.
E = Ointegral(E dr) = -d/dt integral(integral(B dA))
Michael Faraday developed the world's first electric motor, where a charged wire follows the circular magnetic field created by the electric current.
Electromagnetic induction is how a magnetic field drives an electric current around a circuit.
Magnetic fields apply forces to electric charges, but only if the charges are in motion.
F=qV x B
The direction of the force is perpendicular to the velocity and the field, and depends on the sign of the charge (negative in this case). Moving a wire charged with electrons through a magnetic field causes the current to flow.
The field of a bar magnet can be perpendicular to every point of a circular loop of wire. Moving the loop drives a current around the loop. Moving the loop up drives the current one way, and moving the loop down drives the current another way. This also applies if the magnet is moved.
Any method of changing magnetic field makes current flow. This causes a change in magnetic flux which changes the current flow as if by voltage. E = d (IO) or Faraday's law. of electro magnetic induction.
Continually rotating a loop of wire in a magnetic field changes the current in the loop. The resulting voltage and current are sinusoidal, going first one way and then the other.
Current flows through a coil of wire when the magnetic flux through it is changing, the current flows one way if the flux decreases and the other way when the flux increases. The current induced in the wire creates a flux of its own, whose direction depends on the dircetion of the current. The flux created by the induced current always opposes a change in the external flux. This is known as Lenz'z law.
An electric current in any circuit creates a magnetic field which whenever it changes induces a current in the same circuit that opposes changes. This is called self-induction.
The induction of a circuit element:
E = -L dI/dt L= (dIO)/dI
Both resstors and capacitor exhibit self-induction. Electric fields will only circulate if magnetic flux is changed.
E = Ointegral(E dr) = -d/dt integral(integral(B dA))
Sunday, April 1, 2012
Lesson 36 Vector Fields and Hydrodynamics
A disturbance in a field propagates at the speed of light. The study of hydrodynamics led to theories regarding fields of force. The fields of electricity, magnetism, and water are all similar in nature.
The flow of water can be modeled by vectors. Electric forces on a test charge can be represened by a pattern of vectors in space. So can the magnetized needle and its direction in space.
0integral(integral( B dA)) = 0
0integral(integral( E dA)) = a/E(o)
0integral( B dr) = u(o) I
0integral( E dr) = 0
All vector fields obey equations describing their forms. The electric flux through any closed surface is proportional to the net electric charge inside. Magnetic flux through any closed surface is equal to zero. The line integral of any electric field of a closed path is zero. The line integral of the magnetic field around a closed path is proportional to the electric current passing through the path.
In terms of hydrodynamics, flux is the total amount of water passing through an element of are with a given amount of time. In time t that's the total amount of water in a box of width vt where
volume = vtA
Therefore the flux of flow rate is volume/t = vA
If the area element is tilted with respect to the flow, less of the water passes through it. Flux depends on the angle between the flow field and the area element. This idea is similar where both magnetic and electric flux are measured by the total number of lines of force passing through a surface in a given area.
Therefore the mathematical expression of flux is the same for all vector fields.
In water, the flux out of a closed surface is zero where
0integral(integral(V dA)) = 0
Electric flux is like the flux of sunlight. The flux of light as it radiates outward is proportional to the strength of the source within the closed surface.
If there is no flux in magnetic and water fields, how do they have flow?
Water:
If water is stirred motion starts from the outside and moves inward. If water circulates, it does so in a vortex flow that is stable. mRV = L or mass times distance from the center times the speed is conserved for each bit of water.
As each bit of water flows closer to the center hole, speed increases as a vortex is formed because angular momentum is conserved.
V = L/m (1/R) 0^
In magnetism, the vectors represent the strength of the field and not speed.
0integral(V dr) = K
The line integral of the velocity field of water is called circulation. Since the water flows in circles with velocity inversely proportional to distance from the core (the circulation around the vortex core) doesn't depend on distance from the core, it is a constant.
In the same way the line integral of a magnetic field is proportional to the electric current that creates the field. The vortex is the source of fluid flow in the same sense that electric current is the source of the magnetic field. Just as electric current makes a closed loop, so can a vortex core form a loop called a vortex ring. The flow field of a vortex ring is the same as the magnetic field of a current loop. And vortex rings like all vortices are stable structures.
Electric, magnetic, or water, each can be expressed through the line integral and flux.
Magnetic fields circulate but never converge to a point, electric fields radiate from point charges but never circulate. Vector fields also have energy. Energy resides in the motion of the fluid, it is kinetic in nature where
Energy/Volume = 1/2(density) v^2
The energy at a point is proportional to the square of the field E, or B.
The flow of water can be modeled by vectors. Electric forces on a test charge can be represened by a pattern of vectors in space. So can the magnetized needle and its direction in space.
0integral(integral( B dA)) = 0
0integral(integral( E dA)) = a/E(o)
0integral( B dr) = u(o) I
0integral( E dr) = 0
All vector fields obey equations describing their forms. The electric flux through any closed surface is proportional to the net electric charge inside. Magnetic flux through any closed surface is equal to zero. The line integral of any electric field of a closed path is zero. The line integral of the magnetic field around a closed path is proportional to the electric current passing through the path.
In terms of hydrodynamics, flux is the total amount of water passing through an element of are with a given amount of time. In time t that's the total amount of water in a box of width vt where
volume = vtA
Therefore the flux of flow rate is volume/t = vA
If the area element is tilted with respect to the flow, less of the water passes through it. Flux depends on the angle between the flow field and the area element. This idea is similar where both magnetic and electric flux are measured by the total number of lines of force passing through a surface in a given area.
Therefore the mathematical expression of flux is the same for all vector fields.
In water, the flux out of a closed surface is zero where
0integral(integral(V dA)) = 0
Electric flux is like the flux of sunlight. The flux of light as it radiates outward is proportional to the strength of the source within the closed surface.
If there is no flux in magnetic and water fields, how do they have flow?
Water:
If water is stirred motion starts from the outside and moves inward. If water circulates, it does so in a vortex flow that is stable. mRV = L or mass times distance from the center times the speed is conserved for each bit of water.
As each bit of water flows closer to the center hole, speed increases as a vortex is formed because angular momentum is conserved.
V = L/m (1/R) 0^
In magnetism, the vectors represent the strength of the field and not speed.
0integral(V dr) = K
The line integral of the velocity field of water is called circulation. Since the water flows in circles with velocity inversely proportional to distance from the core (the circulation around the vortex core) doesn't depend on distance from the core, it is a constant.
In the same way the line integral of a magnetic field is proportional to the electric current that creates the field. The vortex is the source of fluid flow in the same sense that electric current is the source of the magnetic field. Just as electric current makes a closed loop, so can a vortex core form a loop called a vortex ring. The flow field of a vortex ring is the same as the magnetic field of a current loop. And vortex rings like all vortices are stable structures.
Electric, magnetic, or water, each can be expressed through the line integral and flux.
Magnetic fields circulate but never converge to a point, electric fields radiate from point charges but never circulate. Vector fields also have energy. Energy resides in the motion of the fluid, it is kinetic in nature where
Energy/Volume = 1/2(density) v^2
The energy at a point is proportional to the square of the field E, or B.
Lesson 35 The Magnetic Field
Electric currents produce magnetic forces that cause magnets to point perpendicular to the flow of the current. The electric current that flows through a wire creates a magnetic field that circulates around the wire. The strength of the field depends on the distance from the wire where
delta (B) is proportional to K(m) i/r^2 delta (S) r^
a segment of electric current produces a magnetic field proportional to the inverse square of the distance. The direction of the field depends on the direction of the current according to the vector cross product. The field would be largest where the current segment and distance vector are perpendicular. Electric currents cannot exist in tiny segments so B = K(m) I integral (dsr^/r^2).
B = K(m) 2I/R s^ x R^, the field due to current flowing in a long straight wire is always perpendicular to the wire and decreases as the inverse first power of the distance from the wire. The field is in circles concentric to the wire. If the circular field is placed in a loop, a dipole field forms. A solenoid is a stack of current loops and creates a field much like that of a bar magnet. If the solenoid is bent into a circle, it is called a toroid, in which the magnetic field is contained.
Electric charges apply forces of magnetism between each other. The force of electric current is measured by the force between two wires, or amp.
Magnetism is electricity in motion. Ampere theorized that every magnet must have circulating electric current to produce a magnetic field (Electrodynamics).
In an electric field no work is done is a charge is moved in a closed path, therefore the electric potential is constant and the force of the electric field is zero.
The current of a wire creates circles of constant magnetic field. Since the field is constant on each circle, the line integral on each one is easily calculated where
0integral (B dr) = m 2I/R 2piR 0integral or a constant multiplied by the current in the wire.
0integral (B dr) = u(o) I which is the same for any circle around the wire.
This is Ampere's Law:
0integral(integral( B dA)) = 0
0integral(integral( E dA)) = a/E(o)
0integral( B dr) = u(o) I
0integral( E dr) = 0
The law of electricity and magnetism.
delta (B) is proportional to K(m) i/r^2 delta (S) r^
a segment of electric current produces a magnetic field proportional to the inverse square of the distance. The direction of the field depends on the direction of the current according to the vector cross product. The field would be largest where the current segment and distance vector are perpendicular. Electric currents cannot exist in tiny segments so B = K(m) I integral (dsr^/r^2).
B = K(m) 2I/R s^ x R^, the field due to current flowing in a long straight wire is always perpendicular to the wire and decreases as the inverse first power of the distance from the wire. The field is in circles concentric to the wire. If the circular field is placed in a loop, a dipole field forms. A solenoid is a stack of current loops and creates a field much like that of a bar magnet. If the solenoid is bent into a circle, it is called a toroid, in which the magnetic field is contained.
Electric charges apply forces of magnetism between each other. The force of electric current is measured by the force between two wires, or amp.
Magnetism is electricity in motion. Ampere theorized that every magnet must have circulating electric current to produce a magnetic field (Electrodynamics).
In an electric field no work is done is a charge is moved in a closed path, therefore the electric potential is constant and the force of the electric field is zero.
The current of a wire creates circles of constant magnetic field. Since the field is constant on each circle, the line integral on each one is easily calculated where
0integral (B dr) = m 2I/R 2piR 0integral or a constant multiplied by the current in the wire.
0integral (B dr) = u(o) I which is the same for any circle around the wire.
This is Ampere's Law:
0integral(integral( B dA)) = 0
0integral(integral( E dA)) = a/E(o)
0integral( B dr) = u(o) I
0integral( E dr) = 0
The law of electricity and magnetism.
Lesson 34 Magnetism
William Gilbert discovered that one can destroy the magnetic properties of a metal by heating it up. One can increase magnetic power by stroking one metal with another. The earth behaves like a giant magnet. If you keep an iron bar strictly aligned for a long period of time, it will become magnetized.
Stars and planets have magnetic fields.
The equation for the force between two magnetic poles is
F(m) = K(m) (p1p2/r^2) r^
where opposite poles attract and like poles repel.
Unlike electric charges, magnetic poles always come in equal and opposite pairs. Cutting a magnet in half creates two new poles with a north and a south pole.
In the vast universe, some magnetic mono-poles do exist as a result of the Big Bang.
A magnetic field of a magnet with two poles is similar to the electric field of electric charges with equal and opposite charges.
The circular loop of electric current creates a magnetic field of this form. So do all protons, neutrons and electrons. The earth itself has a dipole field that points South, which is why compass needles point North.
In any magnetic field a magnet is subject to equal and opposite forces at its poles, so it tends to line up with the field. The field exerts a torque that makes the north pole point in the direction of the field.
The Earth's magnetic field is impacted by solar winds, thus not extending indefinitely in the solar direction. Earth's tail, much like a comet's, is comprised of magnetic flux.
Magnetic flux is defined in perfect analogy to electric flux, or the flow of the field through any surface.
d I(O) (E) = E dA
Electric flux through a small element of surface is equal to the area multiplied by the component of the electric field perpendicular to it.
The total flux is the sum of all the flux through the surface.
I(O) (E) = integral(integral(E dA))
= 4piK(E)q
or q/epsilon (o)
The flux through any closed surface is equal to a constant times the change inside.
For an electric dipole, Gauss's law applies by balancing outward flux from the positive charge against inward flux from the negative charge. The total flux, like the charge, is zero. Magnetic flux is defined in exactly the same way.
d I(O) (M) = B dA
I(O) (M) = integral(integral(B dA))
= 0
Flux is a measure of the total number of lines of forces passing through any surface.
Since all magnets are dipoles, the total magnetic flux through any closed surface is zero. The outward flux of the north pole and the inward flux of the south pole balance one another.
Any amount of flux put into a magnet releases the same amount of flux (earth). The earth's magnetic field is produced by electric currents due to molten nickel and iron interaction deep beneath the earth's surface.
The magnetic field is continually changing . The earth changes polarity every 500,000 years, where the north and south pole switch roles.
A magnetic field does not apply any force to an electric charge at rest. However, if the electric charge is in motion, the magnetic field applies a force known as the lorentz force where F = qV x B if it is perpendicular both the field and direction of motion of the charge. Since the magnetic force is perpendicular to the velocity, the force does not cause the charge to speed up or slow down. Charges toend to curve around the field in circular or helical paths.
In non-uniform magnetic fields, electric charges can be trapped in the Van Allen radiation belts. Near the polar regions, charged particles get close enough to strike the atmosphere giving off light like the Aurora Borealis at the North and South Pole.
The magnetic field protects the earth from solar flares and winds. Without the magnetic field, life would not be possible on earth.
Stars and planets have magnetic fields.
The equation for the force between two magnetic poles is
F(m) = K(m) (p1p2/r^2) r^
where opposite poles attract and like poles repel.
Unlike electric charges, magnetic poles always come in equal and opposite pairs. Cutting a magnet in half creates two new poles with a north and a south pole.
In the vast universe, some magnetic mono-poles do exist as a result of the Big Bang.
A magnetic field of a magnet with two poles is similar to the electric field of electric charges with equal and opposite charges.
The circular loop of electric current creates a magnetic field of this form. So do all protons, neutrons and electrons. The earth itself has a dipole field that points South, which is why compass needles point North.
In any magnetic field a magnet is subject to equal and opposite forces at its poles, so it tends to line up with the field. The field exerts a torque that makes the north pole point in the direction of the field.
The Earth's magnetic field is impacted by solar winds, thus not extending indefinitely in the solar direction. Earth's tail, much like a comet's, is comprised of magnetic flux.
Magnetic flux is defined in perfect analogy to electric flux, or the flow of the field through any surface.
d I(O) (E) = E dA
Electric flux through a small element of surface is equal to the area multiplied by the component of the electric field perpendicular to it.
The total flux is the sum of all the flux through the surface.
I(O) (E) = integral(integral(E dA))
= 4piK(E)q
or q/epsilon (o)
The flux through any closed surface is equal to a constant times the change inside.
For an electric dipole, Gauss's law applies by balancing outward flux from the positive charge against inward flux from the negative charge. The total flux, like the charge, is zero. Magnetic flux is defined in exactly the same way.
d I(O) (M) = B dA
I(O) (M) = integral(integral(B dA))
= 0
Flux is a measure of the total number of lines of forces passing through any surface.
Since all magnets are dipoles, the total magnetic flux through any closed surface is zero. The outward flux of the north pole and the inward flux of the south pole balance one another.
Any amount of flux put into a magnet releases the same amount of flux (earth). The earth's magnetic field is produced by electric currents due to molten nickel and iron interaction deep beneath the earth's surface.
The magnetic field is continually changing . The earth changes polarity every 500,000 years, where the north and south pole switch roles.
A magnetic field does not apply any force to an electric charge at rest. However, if the electric charge is in motion, the magnetic field applies a force known as the lorentz force where F = qV x B if it is perpendicular both the field and direction of motion of the charge. Since the magnetic force is perpendicular to the velocity, the force does not cause the charge to speed up or slow down. Charges toend to curve around the field in circular or helical paths.
In non-uniform magnetic fields, electric charges can be trapped in the Van Allen radiation belts. Near the polar regions, charged particles get close enough to strike the atmosphere giving off light like the Aurora Borealis at the North and South Pole.
The magnetic field protects the earth from solar flares and winds. Without the magnetic field, life would not be possible on earth.
Lesson 33 Electric Circuits
Controlling the flow of water has allowed civilizations to develop over the course of history. Similarly, electricity is a fluid whose flow can be controlled.
Inventors such as Thomas Edison developed ways to manipulate electricity to illuminate homes and also to produce and distribute electricity through wires and circuits.
The amount of light is determined by the amount of current measured in amps where 1 amp = 1 coulomb/second
Electric current I is the rate of flow of electric charge q, at any instant, the current is the same anywhere along the wire of the circuit. Electric charge is neither created nor destroyed along the way.
I = d/dt
Hans Christian Ørsted used a voltaic pile to deflect a magnetic needle with an electric current and discovered electromagnetism.
Through the use of a voltaic pile, Edison perfected the telegraph, a device where electric current causes a magnet to move in another location, thus enabling long distance communication.
In a telegraph, to prevent signal loss, if the voltage is increased in proportion to the distance.
Ohm's Law - to make a current flow through a conductor a voltage is needed. The current is always proportional to the voltage.
A constant of proportionality is called R or resistance.
V = IR is Ohm's Law
An element with resistance within a current is called a resistor.
Ohm's law does not hold true in all situations, however it is pratical in most.
The amount of electric current that flows through a resistor depends on the voltage drop across it, how wide, how long it is, and what it is made of.
The resistance of an electric resistor is proportional to its length, inversely proportional to its area and proportional to its resistivity to hinder the flow of electrons. This tendency to resist is something all materials have, but to varying degrees. Multiple resistors in a series are called resistor series and is analogous to lengthening the resistor.
Putting resistors side by side increases the area through which electrons can flow (known as resistors in parallel) and have a lower resistance than either one alone.
What resists the flow of electricity in a conductor? In a metal, electrons move constantly in different directions. The electrons orbit with the metal as if it were a molecule. This flow has no resistance and does not create a net flow in or out. If the conductor is in electrostatic equilibrium there is not electric field inside and no voltage difference. If a battery makes electric current flow, equilibrium is destroyed causing an electric field to form within the metal. Inside a perfect crystalline metal, the mobile electrons would continually accelerate. Impurities of crystals cause resistance by preventing acceleration of electrons.
As current flows through a resistor the energy that is turned into heat is equal to the amount of charge of flow multiplied by the change in potential. The rate of heating of power consumed is equal to IV. Using Ohm's law, power can be written as P = IV = I^2R or V^2/R
1 watt = 1 amp x a volt
Conservation of charge and of energy are derived from Ohm's law.
A capacitor in a circuit stores charge.
Time is equal to capacitance, or resistance for capacitor to empty.
Inventors such as Thomas Edison developed ways to manipulate electricity to illuminate homes and also to produce and distribute electricity through wires and circuits.
The amount of light is determined by the amount of current measured in amps where 1 amp = 1 coulomb/second
Electric current I is the rate of flow of electric charge q, at any instant, the current is the same anywhere along the wire of the circuit. Electric charge is neither created nor destroyed along the way.
I = d/dt
Hans Christian Ørsted used a voltaic pile to deflect a magnetic needle with an electric current and discovered electromagnetism.
Through the use of a voltaic pile, Edison perfected the telegraph, a device where electric current causes a magnet to move in another location, thus enabling long distance communication.
In a telegraph, to prevent signal loss, if the voltage is increased in proportion to the distance.
Ohm's Law - to make a current flow through a conductor a voltage is needed. The current is always proportional to the voltage.
A constant of proportionality is called R or resistance.
V = IR is Ohm's Law
An element with resistance within a current is called a resistor.
Ohm's law does not hold true in all situations, however it is pratical in most.
The amount of electric current that flows through a resistor depends on the voltage drop across it, how wide, how long it is, and what it is made of.
The resistance of an electric resistor is proportional to its length, inversely proportional to its area and proportional to its resistivity to hinder the flow of electrons. This tendency to resist is something all materials have, but to varying degrees. Multiple resistors in a series are called resistor series and is analogous to lengthening the resistor.
Putting resistors side by side increases the area through which electrons can flow (known as resistors in parallel) and have a lower resistance than either one alone.
What resists the flow of electricity in a conductor? In a metal, electrons move constantly in different directions. The electrons orbit with the metal as if it were a molecule. This flow has no resistance and does not create a net flow in or out. If the conductor is in electrostatic equilibrium there is not electric field inside and no voltage difference. If a battery makes electric current flow, equilibrium is destroyed causing an electric field to form within the metal. Inside a perfect crystalline metal, the mobile electrons would continually accelerate. Impurities of crystals cause resistance by preventing acceleration of electrons.
As current flows through a resistor the energy that is turned into heat is equal to the amount of charge of flow multiplied by the change in potential. The rate of heating of power consumed is equal to IV. Using Ohm's law, power can be written as P = IV = I^2R or V^2/R
1 watt = 1 amp x a volt
Conservation of charge and of energy are derived from Ohm's law.
A capacitor in a circuit stores charge.
Time is equal to capacitance, or resistance for capacitor to empty.
Lesson 32 The Electric Battery
To understand a battery, understand the process of making a metal. Begin with a positive ion, because it has a positive charge, it creates the potential energy of attraction for the missing electron. Place positive ions adjacent to one another. Add just enough electrons to make a neutral system and arrange in the form of a lattice to create a piece of metal. A positive test charge would detect no change in potential energy and therefore no force anywhere inside or outside the metal. A real electron inside the metal is acted on by all the ions and electrons except for itself, lending itself a net potential energy. If the electron is moved from one part of the metal to another, other electrons flow to replace it so it has the same potential energy everywhere. There is virtually no force preventing an electron from moving freely through the metal. Beyond the surface of the metal, there are no more ions or electrons to balance forces. So to push an electron outside of metal requires a powerful force, creating a real electric potential and leaving a net positive charge on the metal. The overall potential energy of a real electron drops sharply at the metals surface to a lower value that is the same everywhere inside.
The electrons of a metal also have kinetic energy, but not enough to escape. The amount of energy an electron would need to escape from a metal is known as the work function.
Volta developed the perpetual resevoir of electricity, or the electrophorous.
Galvani studied "animal electricity" using frog legs to which he discovered that nerve impulses which excite muscles into action are really electrical sigals which travel everywhere throughout the body.
Why is an electric impulse created when one metal touches another?
Each metal has a work function that keeps electrons from escaping. The work function of copper and zinc differ from one another by the electric potential IV. When two metals are brought into contact the barrier at the interface vanishes. The electrons are free to flow into the metal where they have lowered total energy. As electrons flow, the metal they leave becomes positively charged and the one they enter becomes negatively charged. This creates an electrostatic potential difference that balance the energy difference. The flow stops. If the metals are separated, each has a net electric charge and an electrostatic potential difference between them.
Volta did not have the luxury of the electron, so he developed the voltaic pile.
Suppose two different metals electrically charged from contact are placed in an electrolite, the metal has extra electrons, attracts positive ions from the solution whenever positive ions touch the surface they can extract excess electrons from the metal. Meanwhle, the other metal which lacks electrons attracts negative ions when negative ions reach the surface, the missing electrons can be replaced. So the electrolite drives the metals back to an electrically neutral position. If they were to make direct contact, a new surge of electrons would flow because of their difference in work fucnctions. Now, because of the electrolite, the surge does not cease. Electrons continue to flow from one metal to the other and they are continually replaced by the ions in the solution. Until there is no more chemical energy. This is known as the battery.
The electrons of a metal also have kinetic energy, but not enough to escape. The amount of energy an electron would need to escape from a metal is known as the work function.
Volta developed the perpetual resevoir of electricity, or the electrophorous.
Galvani studied "animal electricity" using frog legs to which he discovered that nerve impulses which excite muscles into action are really electrical sigals which travel everywhere throughout the body.
Why is an electric impulse created when one metal touches another?
Each metal has a work function that keeps electrons from escaping. The work function of copper and zinc differ from one another by the electric potential IV. When two metals are brought into contact the barrier at the interface vanishes. The electrons are free to flow into the metal where they have lowered total energy. As electrons flow, the metal they leave becomes positively charged and the one they enter becomes negatively charged. This creates an electrostatic potential difference that balance the energy difference. The flow stops. If the metals are separated, each has a net electric charge and an electrostatic potential difference between them.
Volta did not have the luxury of the electron, so he developed the voltaic pile.
Suppose two different metals electrically charged from contact are placed in an electrolite, the metal has extra electrons, attracts positive ions from the solution whenever positive ions touch the surface they can extract excess electrons from the metal. Meanwhle, the other metal which lacks electrons attracts negative ions when negative ions reach the surface, the missing electrons can be replaced. So the electrolite drives the metals back to an electrically neutral position. If they were to make direct contact, a new surge of electrons would flow because of their difference in work fucnctions. Now, because of the electrolite, the surge does not cease. Electrons continue to flow from one metal to the other and they are continually replaced by the ions in the solution. Until there is no more chemical energy. This is known as the battery.
Lesson 31 Voltage Energy and Force
Voltage (electric potential) is a measure of the electric charge. The following observation unveils the dilemma between electric force and potential. Electrons in a body are held together by only 3-5v, and batteries cannot shoot electric beams like the Van de Graaff generator.
The electric field is the negative derivative of potential.
v = -integral (E dr)
E = K(e) (q/r^2) r^
V = -K(e) integral (q/r^2 dr)
V = K(e) q/r
The integral of the electric field of a point charge is proportional to 1/r.
It requires no work to move a charge along a curve of constant potential.
The electric field is perpendicular to each equal potential at every point. Electric potential is the ability to do work by making electric charges flow, this potential is measured in volts.
The electric chair was one of the first uses of high voltage (neon lights)
Atoms, the basis of all matter is held together by electricity.
In every atom, the electric force binds negatively charged electrons to a positive nucleus. The nucleus can be considered a point charge even though it is comprised of protons and neutrons. One electron has exactly the amount of negative charge to balance with the positive charge, and result in a perfectly neutral atom.
The distance between the nucleus and the outermost electron is 1 Angstrom = 10^-8 cm.
All other electrons balance with all protons except one, and the remaining electron detects the electric field of the proton only 1 A(o) away.
The electric potential = 14.4 V
U = qV
because charge is negative, so is potential energy at 14.4eV, this must be overcome to remove an electron from an atom:
If an atom and a 100,000 V vandegraph fight over an electron the atom will win. This is not
because of voltage, but force. The derivative of potential energy.
The force of the atom is 100,000 times stronger than that of the Van de Graaff.
In lightning, the molecules of air are momentarily ionized leaving a gas of positive molecular ions and negative electrons called a plasma. The electric force between ions and electrons causes them to recombine into neutral matter giving off excess energy in the form of light.
In a neon light, the process is slowed to a continuous glow, but the electric field in a neon light cannot ionize matter.
The neon light and Van de Graaff generator ionize air through collisions that contains not electric force, but a few accidental electrons that hit other molecules. If the electric field is great enough, thus increasing acceleration and the distance between atoms is large enough, the electron can build enough energy to knock another electron off an atom. This causes a chain reaction due to higher acceleration and a spark.
The force of electricity depends on both voltage and charge. Voltage, energy, and force hold the universe together.
The electric field is the negative derivative of potential.
v = -integral (E dr)
E = K(e) (q/r^2) r^
V = -K(e) integral (q/r^2 dr)
V = K(e) q/r
The integral of the electric field of a point charge is proportional to 1/r.
It requires no work to move a charge along a curve of constant potential.
The electric field is perpendicular to each equal potential at every point. Electric potential is the ability to do work by making electric charges flow, this potential is measured in volts.
The electric chair was one of the first uses of high voltage (neon lights)
Atoms, the basis of all matter is held together by electricity.
In every atom, the electric force binds negatively charged electrons to a positive nucleus. The nucleus can be considered a point charge even though it is comprised of protons and neutrons. One electron has exactly the amount of negative charge to balance with the positive charge, and result in a perfectly neutral atom.
The distance between the nucleus and the outermost electron is 1 Angstrom = 10^-8 cm.
All other electrons balance with all protons except one, and the remaining electron detects the electric field of the proton only 1 A(o) away.
The electric potential = 14.4 V
U = qV
because charge is negative, so is potential energy at 14.4eV, this must be overcome to remove an electron from an atom:
If an atom and a 100,000 V vandegraph fight over an electron the atom will win. This is not
because of voltage, but force. The derivative of potential energy.
The force of the atom is 100,000 times stronger than that of the Van de Graaff.
In lightning, the molecules of air are momentarily ionized leaving a gas of positive molecular ions and negative electrons called a plasma. The electric force between ions and electrons causes them to recombine into neutral matter giving off excess energy in the form of light.
In a neon light, the process is slowed to a continuous glow, but the electric field in a neon light cannot ionize matter.
The neon light and Van de Graaff generator ionize air through collisions that contains not electric force, but a few accidental electrons that hit other molecules. If the electric field is great enough, thus increasing acceleration and the distance between atoms is large enough, the electron can build enough energy to knock another electron off an atom. This causes a chain reaction due to higher acceleration and a spark.
The force of electricity depends on both voltage and charge. Voltage, energy, and force hold the universe together.
Lesson 30 Capacitance and Potential
In 1745, Pieter van Musschenbroek, wanted to make an electric field solution and developed the Leyden jar.
Benjamin Franklin interpreted the Lyden jar and developed an electric theory concerning charge.
In the vacinity of a positive charge, the electric force repels a positive test charge, and an external force is needed to push the charge closer doing work against the electrical force, a positive force (work) if the component does work that is opposite to the electric force.
dW = -F dr
It is known as negative work if it has a component along the force and no work at all if the motion is perpendicular to the electric force.
The net work is found by
integral (dw) = integral (-F dr)
delta (w) = -integral (F dr)
The net work is delta (U) or the change in potential energy of the test charge.
delta (U) = -integral ( F dr)
F = qE
The electric force is the charge multiplied by the electric field.
delta (U) = q delta (V)
delta (V) = -integral (E dr) or the change in electric potential concerning only the path of the charge through the field. it is measured in volts.
Before the electric field, Franklin theorized the electric atmosphere where all objects contain electrical fluid, where objects with too much fluid have a positive charge and objects with too little fluid have a negative charge. Franklin understood that electric charge in never created nor destroyed, but flows from one object to the next.
All objects with a net electrical charge produce an electricl field. If objects contain both positive and negative charges, the electric field exists, but is small in nature, and renders the object with a neutral charge. Inside metals, there is not static electricity because all the charged atoms of the metal are attracted to the surface and prevent further movement.
positive charge: >U
negative charge: <U
where q is proportional to V
A battery can create an electric field where -- as electricity flows between two conductors -- the difference in potential energy is equivalent to the voltage of the battery, and creates an electric field.
q = CV
The charge transfered is proportional to the voltage applied, and the constant of proportionality is C (capacitance).
Franklin revolutionized physics by discovering that electric force is neither created nor destroyed, but transfered from source to source according to the electric charge.
A capacitor can be made with any two pieces of metal. A parallel plate capacitor with two sheets of opposite charge create an electric field between themselves, with the total amount of voltage proportional to distance between the two plates. A parallel plate capacitor was present within Leyden jars.
Benjamin Franklin interpreted the Lyden jar and developed an electric theory concerning charge.
In the vacinity of a positive charge, the electric force repels a positive test charge, and an external force is needed to push the charge closer doing work against the electrical force, a positive force (work) if the component does work that is opposite to the electric force.
dW = -F dr
It is known as negative work if it has a component along the force and no work at all if the motion is perpendicular to the electric force.
The net work is found by
integral (dw) = integral (-F dr)
delta (w) = -integral (F dr)
The net work is delta (U) or the change in potential energy of the test charge.
delta (U) = -integral ( F dr)
F = qE
The electric force is the charge multiplied by the electric field.
delta (U) = q delta (V)
delta (V) = -integral (E dr) or the change in electric potential concerning only the path of the charge through the field. it is measured in volts.
Before the electric field, Franklin theorized the electric atmosphere where all objects contain electrical fluid, where objects with too much fluid have a positive charge and objects with too little fluid have a negative charge. Franklin understood that electric charge in never created nor destroyed, but flows from one object to the next.
All objects with a net electrical charge produce an electricl field. If objects contain both positive and negative charges, the electric field exists, but is small in nature, and renders the object with a neutral charge. Inside metals, there is not static electricity because all the charged atoms of the metal are attracted to the surface and prevent further movement.
positive charge: >U
negative charge: <U
where q is proportional to V
A battery can create an electric field where -- as electricity flows between two conductors -- the difference in potential energy is equivalent to the voltage of the battery, and creates an electric field.
q = CV
The charge transfered is proportional to the voltage applied, and the constant of proportionality is C (capacitance).
Franklin revolutionized physics by discovering that electric force is neither created nor destroyed, but transfered from source to source according to the electric charge.
A capacitor can be made with any two pieces of metal. A parallel plate capacitor with two sheets of opposite charge create an electric field between themselves, with the total amount of voltage proportional to distance between the two plates. A parallel plate capacitor was present within Leyden jars.
Lesson 29 The Electric Field
Michael Faraday developed the idea of the electric field as lines of constant electric force radiating everywhere throughout space (known as the field theory).
Charles Augustine Coulomb:
F(e) = K(e) (q1q2/r^2) r^
where the electric force is inversely proportional to the square of the distance between two charges.
The Universal Law of Gravitation also follows a similar principle where F(g) = -G (m1m2/r^2)r^. This law led to the development of the theory and phrase "action at a distance" where bodies such as the earth and sun directly apply force to one another over copius kilometers.
F(e) = K(e) (q1q2/r^2) r^
F(g) = -G (m1m2/r^2)r^
F(m) = K(m) (p1p2/r^2)r^
All these laws have a force that decreases with the square of the distance.
The inverse square law is related to a simple geometric characteristic of space, known as flux, where intensity alpha = 1/r^2 describing the radiating light of the sun.
Faraday set out to solve the scientific mystery of why a compass needle spins to a perpendicular position from an electric charge, and developed an electric motor.
Anywhere in the vacinity of an electric charge a small test charge experiences a force. If it is due to only one charge the pattern of forces detected by the test charge is simple where similar forces repel and opposite forces attract.
The pattern of forces is present in a space as a field and can be expressed mathematically
F = sum (K(e) (qqi/ri^2) r^i)
The force that acts on a test charge at each point in space is equal to the test charge times a quantity of the other charges. That quantity is the electric field F = qE E= sum (K(e) (qqi/ri^2) r^i)
The 1/r^2 force between electric charges suggest that the force must be applied by something radiating outward from charges, something which like light from the sun never stops and never ends in space. These forces take characteristics of lines that never cross or angle. This electric field force is stronger near charges where lines (vectors) are close together and weak where the lines are far apart.
Gauss developed the mathematics of the Electric field theory through Gauss's Law:
for any closed surface total flux is proportional to the net electric charge inside. If there's no net charge inside a surface, any positive flux outward through it, must be balanced by inward, negative flux.
This law applies to light, gravitational fields, magnetic fields, and electric fields.
An electric field passing through a conductor forces the electrons to flow until they pile up at the surface repelling further motion of electron. The electric fiel inside any conductor becomes equal to zero when electrostatic equilibrium is established. Therefore, a closed surface inside the conductor has no flux through it, so the net charge inside must be zero, bu there can be charge at the surface. No matter what is outside, the surface charge makes the electric field inside equal to zero.
A metal box of any kind can keep out an electric field, known as the Faraday cage. This explains why drivers lose radio reception when passing through tunnels or bridges.
Gauss's law proves the theory of the center of mass.
Maxwell worked to express the electric field in mathematical terms.
Charles Augustine Coulomb:
F(e) = K(e) (q1q2/r^2) r^
where the electric force is inversely proportional to the square of the distance between two charges.
The Universal Law of Gravitation also follows a similar principle where F(g) = -G (m1m2/r^2)r^. This law led to the development of the theory and phrase "action at a distance" where bodies such as the earth and sun directly apply force to one another over copius kilometers.
F(e) = K(e) (q1q2/r^2) r^
F(g) = -G (m1m2/r^2)r^
F(m) = K(m) (p1p2/r^2)r^
All these laws have a force that decreases with the square of the distance.
The inverse square law is related to a simple geometric characteristic of space, known as flux, where intensity alpha = 1/r^2 describing the radiating light of the sun.
Faraday set out to solve the scientific mystery of why a compass needle spins to a perpendicular position from an electric charge, and developed an electric motor.
Anywhere in the vacinity of an electric charge a small test charge experiences a force. If it is due to only one charge the pattern of forces detected by the test charge is simple where similar forces repel and opposite forces attract.
The pattern of forces is present in a space as a field and can be expressed mathematically
F = sum (K(e) (qqi/ri^2) r^i)
The force that acts on a test charge at each point in space is equal to the test charge times a quantity of the other charges. That quantity is the electric field F = qE E= sum (K(e) (qqi/ri^2) r^i)
The 1/r^2 force between electric charges suggest that the force must be applied by something radiating outward from charges, something which like light from the sun never stops and never ends in space. These forces take characteristics of lines that never cross or angle. This electric field force is stronger near charges where lines (vectors) are close together and weak where the lines are far apart.
Gauss developed the mathematics of the Electric field theory through Gauss's Law:
for any closed surface total flux is proportional to the net electric charge inside. If there's no net charge inside a surface, any positive flux outward through it, must be balanced by inward, negative flux.
This law applies to light, gravitational fields, magnetic fields, and electric fields.
An electric field passing through a conductor forces the electrons to flow until they pile up at the surface repelling further motion of electron. The electric fiel inside any conductor becomes equal to zero when electrostatic equilibrium is established. Therefore, a closed surface inside the conductor has no flux through it, so the net charge inside must be zero, bu there can be charge at the surface. No matter what is outside, the surface charge makes the electric field inside equal to zero.
A metal box of any kind can keep out an electric field, known as the Faraday cage. This explains why drivers lose radio reception when passing through tunnels or bridges.
Gauss's law proves the theory of the center of mass.
Maxwell worked to express the electric field in mathematical terms.
Saturday, March 17, 2012
Lesson 28 Static Electricity
In the 1700s science turned towards the use of electricty after Franklin determined that "charge" is the source of electrical force. French physicist Coulomb found the relation between charge and force where F(electricty) = k(electricty) q1 q2/(r^2) r^
Electrical force is proportional to the product of the charges and inversely proportional to the square of the distance between the two charges.
F(electricty) = + r^ - repel
F(electricity) = - r^ - negative
The force two charges exert on a third is the vector sum of the forces each alone would exert. In fact, any number of charges, either positive or negative, have a total force of a vector sum
F = sum( ke qqi/ri^2 r^i)
Electric charge exerts electric force and obeys Coulomb's law of magnitude and sign of the electrostatic force between two idealized point charges.
Electricity resides in matter, solids, liquids, and gasses of the universe. Matter is electrical in nature as it is held together by the interaction between positive and negative forces.
Electricity is a fluid comprised of atoms held together by positive and negative charges.
At the core of every atom is a nucleus containing protons, neutrons, and electrons. Protons and electrons create a balance net electrical force that is neutral.
Ions are atoms that contain either too many protons or too many electrons.
Metals exhibit the properties of electricity, are malleable, ductile, and are capable of changing shape without breaking. Metals also have properties of luster and conducivity.
On an insulator, an electric charge stays where it is located. However, with a conductor the charge spreads throughout an object. This is because metals can be thought of as positive ions with loosely bound electrons. When two metallic ions are next to each other the electrons can pass easily from one ion to the next. The mobile electrons are known as conductor electrons and give metals their properties.
The Van de Graff generator is an electrostatic generator that uses a moving belt to create high voltage. However, the voltage of the generator may be larger than the voltage of lightning created by friction between ice particles in clowds, but due to capacitance, lighting is extremely destructive.
Electricity was first discovered through the use of Leyden jars or primative batteries. The electric capabilities of modern bateries are hundreds of thousads stronger than that those of Leyden jars.
Electricity resides in matter, solids, liquids, and gasses of the universe
Electrical force is proportional to the product of the charges and inversely proportional to the square of the distance between the two charges.
F(electricty) = + r^ - repel
F(electricity) = - r^ - negative
The force two charges exert on a third is the vector sum of the forces each alone would exert. In fact, any number of charges, either positive or negative, have a total force of a vector sum
F = sum( ke qqi/ri^2 r^i)
Electric charge exerts electric force and obeys Coulomb's law of magnitude and sign of the electrostatic force between two idealized point charges.
Electricity resides in matter, solids, liquids, and gasses of the universe. Matter is electrical in nature as it is held together by the interaction between positive and negative forces.
Electricity is a fluid comprised of atoms held together by positive and negative charges.
At the core of every atom is a nucleus containing protons, neutrons, and electrons. Protons and electrons create a balance net electrical force that is neutral.
Ions are atoms that contain either too many protons or too many electrons.
Metals exhibit the properties of electricity, are malleable, ductile, and are capable of changing shape without breaking. Metals also have properties of luster and conducivity.
On an insulator, an electric charge stays where it is located. However, with a conductor the charge spreads throughout an object. This is because metals can be thought of as positive ions with loosely bound electrons. When two metallic ions are next to each other the electrons can pass easily from one ion to the next. The mobile electrons are known as conductor electrons and give metals their properties.
The Van de Graff generator is an electrostatic generator that uses a moving belt to create high voltage. However, the voltage of the generator may be larger than the voltage of lightning created by friction between ice particles in clowds, but due to capacitance, lighting is extremely destructive.
Electricity was first discovered through the use of Leyden jars or primative batteries. The electric capabilities of modern bateries are hundreds of thousads stronger than that those of Leyden jars.
Electricity resides in matter, solids, liquids, and gasses of the universe
Lesson 27 Beyond the Mechanical Universe
Levi-Civita and Einstein exchanged letters concerning the Theory of Relativity because Levi-Civita had a strong basis in mathematics while Einstein has a basis in Physics. Levi-Civita found errors in Einstein's theory concerning tensors, or a generalized vector describing the magnetic field, and how those tensors change from coordinate system to coordinate system.
On the other hand, Franklin and Faraday revolutionized the sciences and paved the way for physics and many theories. Faraday discovered that like gravity, electricity and magnetism decrease with the square of the distance. Faraday also found tht any forces described by (1/r^2) must radiate outward. These forces repel and attract, and became known as electric and magnetic fields.
James Clark Maxwell built upon Faraday's discoveries and developed the electromagnetic field theory:
1.) integral(integral(E dA)) = q/E(o)
2.) integral(integral( B dA)) = 0
3.) integral(E dr) = -d (Phi)/dt
4.) integral(B dr) = Mu(o) (I + (Epsilon)(o) d(Phi)/dt)
All of these ideas flourished during the Industrial Revolution. Franklin expanded upon the field of electricity through the use of Leyden jars and found that a positive and negative electrical force will attract each other, and if the two forces are positive they will repel each other. Franklin developed the terms "positive charge" and "negative charge."
Luigi Galvani was the world's first neurobiologist and studied frog's legs and their response to an electrical charge.
As other resources faded out during the Industrial Revolution, electricity slowly achieved dominance.
Faraday discovered electromagnetic induction. Electromagnetic induction states that by increasing or decreasing the current in one electric circuit the changing magnetic field induced a current to flow in a second circuit.
Thomas Edison soon developed the phonograph, "ticker-tape," and the light bulb. This discovery led to the controversy and question of whether alternating or direct current would propel the world into the future.
Michelson set out to disprove Galileo's theory by discovering the absolute motion of the Earth. Michelson was the first American to win the Nobel Prize, however failed at his experiment because of the fact that regardless of motion, the same speed of light is always observed.
These numerous discoveries and perceptions led into the Theory of Relativity, Quantum Mechanics, past the mechanical universe, and beyond.
On the other hand, Franklin and Faraday revolutionized the sciences and paved the way for physics and many theories. Faraday discovered that like gravity, electricity and magnetism decrease with the square of the distance. Faraday also found tht any forces described by (1/r^2) must radiate outward. These forces repel and attract, and became known as electric and magnetic fields.
James Clark Maxwell built upon Faraday's discoveries and developed the electromagnetic field theory:
1.) integral(integral(E dA)) = q/E(o)
2.) integral(integral( B dA)) = 0
3.) integral(E dr) = -d (Phi)/dt
4.) integral(B dr) = Mu(o) (I + (Epsilon)(o) d(Phi)/dt)
All of these ideas flourished during the Industrial Revolution. Franklin expanded upon the field of electricity through the use of Leyden jars and found that a positive and negative electrical force will attract each other, and if the two forces are positive they will repel each other. Franklin developed the terms "positive charge" and "negative charge."
Luigi Galvani was the world's first neurobiologist and studied frog's legs and their response to an electrical charge.
As other resources faded out during the Industrial Revolution, electricity slowly achieved dominance.
Faraday discovered electromagnetic induction. Electromagnetic induction states that by increasing or decreasing the current in one electric circuit the changing magnetic field induced a current to flow in a second circuit.
Thomas Edison soon developed the phonograph, "ticker-tape," and the light bulb. This discovery led to the controversy and question of whether alternating or direct current would propel the world into the future.
Michelson set out to disprove Galileo's theory by discovering the absolute motion of the Earth. Michelson was the first American to win the Nobel Prize, however failed at his experiment because of the fact that regardless of motion, the same speed of light is always observed.
These numerous discoveries and perceptions led into the Theory of Relativity, Quantum Mechanics, past the mechanical universe, and beyond.
Sunday, January 22, 2012
Lesson 26 Harmony of the Spheres
The heavenly motions are nothing but a continuous song for several voices (perceived by the intellect, not by the ear); a music which, through discordant tensions, through sincopes and cadenzas, as it were (as men employ them in imitation of those natural discords) progresses towards certain pre-designed quasi six-voiced clausuras, and thereby sets landmarks in the immeasurable flow of time. -- The Harmony of the Universe (Harmonice mundi) Johannes Kepler
Heavenly music was an idea that dated back to Pythagoras, and Kepler was the first to record the sounds of the planets in Harmony of the Universe
John Rogers and Willie Ruff of Yale University synthesized Kepler's music which became know as the "Harmony of the Spheres."
The Pythagorean theorem was used to find how fast the moon falls each second, just as the odd numbers add up to the perfect squares.
Studying the musical compositions of the heavens, Europeans developed distinct musical compositions in Europe.
Galileo found that each swing of a pendulum occurs in the same amount of time. This discovery led to keeping perfect time, an undeniably important aspect of music.
Galileo then discovered resonance through the application of a rythmically applied chorus.
Western scholars only made discoveries due to the foundations of science from the Pythagoreans and Greeks that ultimately led to the conclusion that nature is solely based off of mathematics.
Newton's Law F = m a describes all bodies on earth, in the heavens, and throughout the universe. This law united the physics of both earth and space.
All projectiles follow orbits that can be depicted by conic sections due to the fact that F = m a applies to every force in the mechanical universe, and every mass no matter what or where.
F= m a is an equation that relies on derivatives and can describe acceleration, velocity, and position of any projectile ever fired.
Copernicus challenged old theories and needed new mathematics of motion to describe the nature of new discoveries.
Whenever there is no outside force, momentum is constant. Force is the rate of change of momentum. Momentum is conserved and if r x f is 0, angular momentum is constant as long as there is no torque.
Energy is also constant. Work that lifts an object from one height to another becomes potential energy. Work that acclerates a block becomes kinetic energy. The kinetic energy of a falling object is transfered into heat, which leads to the kinetic energy among molecules and atoms.
E = U + K is always constant and determines the orbits of the planets:
e<1 - ellipse
e=1 - parabola
e=0 - circle
e>1 - hyperbola
The harmony of the mechanical universe, containing many mysteries yet to be solved.
Heavenly music was an idea that dated back to Pythagoras, and Kepler was the first to record the sounds of the planets in Harmony of the Universe
John Rogers and Willie Ruff of Yale University synthesized Kepler's music which became know as the "Harmony of the Spheres."
The Pythagorean theorem was used to find how fast the moon falls each second, just as the odd numbers add up to the perfect squares.
Studying the musical compositions of the heavens, Europeans developed distinct musical compositions in Europe.
Galileo found that each swing of a pendulum occurs in the same amount of time. This discovery led to keeping perfect time, an undeniably important aspect of music.
Galileo then discovered resonance through the application of a rythmically applied chorus.
Western scholars only made discoveries due to the foundations of science from the Pythagoreans and Greeks that ultimately led to the conclusion that nature is solely based off of mathematics.
Newton's Law F = m a describes all bodies on earth, in the heavens, and throughout the universe. This law united the physics of both earth and space.
All projectiles follow orbits that can be depicted by conic sections due to the fact that F = m a applies to every force in the mechanical universe, and every mass no matter what or where.
F= m a is an equation that relies on derivatives and can describe acceleration, velocity, and position of any projectile ever fired.
Copernicus challenged old theories and needed new mathematics of motion to describe the nature of new discoveries.
Whenever there is no outside force, momentum is constant. Force is the rate of change of momentum. Momentum is conserved and if r x f is 0, angular momentum is constant as long as there is no torque.
Energy is also constant. Work that lifts an object from one height to another becomes potential energy. Work that acclerates a block becomes kinetic energy. The kinetic energy of a falling object is transfered into heat, which leads to the kinetic energy among molecules and atoms.
E = U + K is always constant and determines the orbits of the planets:
e<1 - ellipse
e=1 - parabola
e=0 - circle
e>1 - hyperbola
The harmony of the mechanical universe, containing many mysteries yet to be solved.
Lesson 25 Kepler to Einstein
The periods of the planets in their orbits, the ebb and flow of the tides, the acceleration of a falling body; all these phenomena are consequences of the force of gravity and they have inspired the labor of scientists from Kepler to Einstein.
Einstein was searching for the reason that gravity applies the same force for all bodies to fall at the same rate.
Galileo developed a theory of the tides that explained the tides of the earth. He concluded that it was due to the motion of the earth around the sun. However, this was a problem because for his theory to be true there would need to be one high tide at noon every day. In fact, tides rise and recede twice every day.
In the earth-moon system, the earth and the moon rotate around a common center of mass. This point is actually inside the earth, three-fourths of the distance from the center to the surface. The strange wobbling motion of the earth is a Keplerian orbit of the center of the earth around the center of mass of the earth-moon system. Only the center of the earth is in exactly the rigt orbit. On the side closest to the moon, the moon's gravity is too strong, and the water bulges as it is pulled towards the moon. On the opposite side the moon's gravity is too weak to hold it in place so it bulges as it tries to escape. As the earth wobbles around the earth-moon center of mass, it also rotates on its axis. As the earth rotates it passes beneath the bulges. At those locations where the rotating Earth passes under the rising water, high tides occur. At the points between, low tides occur; two high tides and two low tides. The sun plays a role in the tides at about half of the strength of the moon because of the distance between the earth and the sun. At "new moon" of full moon, the earth, moon, and sun fall in a staight line and the forces of gravity reinforce eachother to create the largest high tides and the smallest low tides.
1st - r = ed/ ecos(theta)+1
Each planet moves in an ellipse with the Sun at one focus.
2nd - dA/dt = constant
A line drawn from the Sun to a planet sweeps out equal areas in equal times.
The square of the period of a planet's orbit is proportional to the cube of the length of the semi - major axis.
Einstein was searching for the reason that gravity applies the same force for all bodies to fall at the same rate.
Galileo developed a theory of the tides that explained the tides of the earth. He concluded that it was due to the motion of the earth around the sun. However, this was a problem because for his theory to be true there would need to be one high tide at noon every day. In fact, tides rise and recede twice every day.
In the earth-moon system, the earth and the moon rotate around a common center of mass. This point is actually inside the earth, three-fourths of the distance from the center to the surface. The strange wobbling motion of the earth is a Keplerian orbit of the center of the earth around the center of mass of the earth-moon system. Only the center of the earth is in exactly the rigt orbit. On the side closest to the moon, the moon's gravity is too strong, and the water bulges as it is pulled towards the moon. On the opposite side the moon's gravity is too weak to hold it in place so it bulges as it tries to escape. As the earth wobbles around the earth-moon center of mass, it also rotates on its axis. As the earth rotates it passes beneath the bulges. At those locations where the rotating Earth passes under the rising water, high tides occur. At the points between, low tides occur; two high tides and two low tides. The sun plays a role in the tides at about half of the strength of the moon because of the distance between the earth and the sun. At "new moon" of full moon, the earth, moon, and sun fall in a staight line and the forces of gravity reinforce eachother to create the largest high tides and the smallest low tides.
Kepler's Three Laws:
2nd - dA/dt = constant
3rd - T^2 = (4pi^2/GM) a^3
dA/dt = L/2M Integrating Kepler's second law through one period, where the period of a planet's orbit is proportional to its area.
T = 2A/L/M holds true for many types of motion
F = -G M Mo/r^2 (r^)
F = m a
a = -D/Mr^2 (r^)
This motion depicts an elliptical orbit whose size depends on L/M ue to Newton's Law of Universal Gravitation.
L^2/DM/(1+ecos(theta))
T = 2A/sqrt(GMoa(1-e^2)) Area = pi a^2 sqrt(1-e^2)
T = 2pi/sqrt(GMo) (a^3) - 4pi^2 is the gravitational constant and the mass of the sun connects T^2 to a^3 for every planet.
T^2 = 4pi^2/GM0 (a^3)
Although this desribes the mechanical properties of the universe, Einstein used gravity, time, an space to solve the deepest mysteries of the cosmos.
Inertial mass and grvitational mass are the same for all bodies in the universe for the Law of Falling Bodies to occur. For this to happen, Einstein searched for a profound law of nature for the law to be veritable. This led to the Theory of Relativity.
The Principle of Equivalence states that there is constant gravity and acceleration that cause bodies to fall at same rate.
The Law of Falling Bodies and the mysterious equality of gravitational mass and inertial mass would be explained if the following fundamental principle were true:
Objects do not fall because of gravity, but appear to fall due to the acceleration of the Earth uppward into space.
Einstein used a beam of light to explain this theory. Einstein stated that in space light would bend downward slightly, and on earth light would bend downward slightly due to the force of the earth's gravity that causes light to not travel in straight lines.
Einstein observed light during an eclipse and was correct about the manner in which light travels.
In theory, there are no straight lines in space. Instead, the shortest distance is known as geodesic on any globe or surface.
Light is bent by the gravity of the sun, but travels inertially along the shortest distance between any two points in local curved space time.
Einstein eliminated gravity and attributed the order of universe to the curved space time. However, how does mass cause space time to curve? Einstein died before this question could be answered, however, under extreme conditions in the universe, only Einstein's Theory of Relativity holds true.
Monday, January 16, 2012
Lesson 24 Navigating in Space
Voyages to other planets require enormous amounts of energy. The amount of energy expended can be minimized using the same force that moves the planets through the solar system. The force of the sun's gravitational field is used to navigate through space.
1973 - Mariner 10 was sent to Venus and Mars
1975 - Viking was sent to Mars
1977 - Voyager was sent to Jupiter, Saturn, Uranus, and Neptune.
For an object to move in a straight line in space it requires a lot of propulsion energy. Instead, scientists recognized Kepler's second law - planets move around the sun in orbitals shaped as ellipses - as an operting principle to allow spacecraft to follow the same path and be projected from around the sun into space.
The spacecraft is put in an independent orbit around the sun that intersects the orbits of Earth and Mars In order for the spacecraft to coast after launch to its destination.
To travel between two points in space the craft coasts to its destination in orbit around the sun, just as if it were an orbiting planet. The path the craft travels from one planet to another is known as the transfer orbit.
The Viking mission was limited in the selection of a transfer orbit because it was the most massive interplanetary craft ever launched. The Viking mission followed a Hohmann transfer, which is the classic method for travel between two planets. The craft was launched when the Earth was at its point in its orbit closest to the Sun and arrived at Mars when Mars was at its point in its orbit farthest from the Sun.
T^2 = 4pi^2/GM (a^3) - Kepler's third law dictates the length of the Earth year and also how long a spacecraft coasting in its own orbit will take to get from the Earth to the position of the orbit of Mars, which is about eight and half months.
A spacecraft must be launched when Mars is 44 degrees ahead of the Earth; this is kown as an opportunity.
Opporunity:
Venus - 54 degrees - every 19 months
Mars - 44 degrees - every 2 years
Jupiter - 97 degrees - every 13 months
There are only certain times each day that a spacecraft can be launched due to the Earth's rotation. Any increase or boost in speed creates a larger more eccentric orbit. To go from the Earth's orbit to a Mars tranfer orbit the craft must be launched at 2.9 km/s. However, to do this the craft must escape the parking orbit around the Earth. A rocket thrust will boost the rocket into a hyperbola orbit at 2.9 km/s; this must occur at 7:56 p.m. The Sun will then bend the hyperbola orbit into a Mars transfer orbit. In order to travel to Venus, the rocket thrust must be activated at 7:40 p.m. in order for the craft to travel inward.
The four outer planets line up for a craft to visit all four every 175 years. The craft must overcome gravity in order to leave the planet and utilizes gravity assist to travel to other planets.
300 years after the discovery of classical mechanics, discoveries are still to be made in space, physics, and every field of science.
1973 - Mariner 10 was sent to Venus and Mars
1975 - Viking was sent to Mars
1977 - Voyager was sent to Jupiter, Saturn, Uranus, and Neptune.
For an object to move in a straight line in space it requires a lot of propulsion energy. Instead, scientists recognized Kepler's second law - planets move around the sun in orbitals shaped as ellipses - as an operting principle to allow spacecraft to follow the same path and be projected from around the sun into space.
The spacecraft is put in an independent orbit around the sun that intersects the orbits of Earth and Mars In order for the spacecraft to coast after launch to its destination.
To travel between two points in space the craft coasts to its destination in orbit around the sun, just as if it were an orbiting planet. The path the craft travels from one planet to another is known as the transfer orbit.
The Viking mission was limited in the selection of a transfer orbit because it was the most massive interplanetary craft ever launched. The Viking mission followed a Hohmann transfer, which is the classic method for travel between two planets. The craft was launched when the Earth was at its point in its orbit closest to the Sun and arrived at Mars when Mars was at its point in its orbit farthest from the Sun.
A spacecraft must be launched when Mars is 44 degrees ahead of the Earth; this is kown as an opportunity.
Opporunity:
Venus - 54 degrees - every 19 months
Mars - 44 degrees - every 2 years
Jupiter - 97 degrees - every 13 months
There are only certain times each day that a spacecraft can be launched due to the Earth's rotation. Any increase or boost in speed creates a larger more eccentric orbit. To go from the Earth's orbit to a Mars tranfer orbit the craft must be launched at 2.9 km/s. However, to do this the craft must escape the parking orbit around the Earth. A rocket thrust will boost the rocket into a hyperbola orbit at 2.9 km/s; this must occur at 7:56 p.m. The Sun will then bend the hyperbola orbit into a Mars transfer orbit. In order to travel to Venus, the rocket thrust must be activated at 7:40 p.m. in order for the craft to travel inward.
The four outer planets line up for a craft to visit all four every 175 years. The craft must overcome gravity in order to leave the planet and utilizes gravity assist to travel to other planets.
300 years after the discovery of classical mechanics, discoveries are still to be made in space, physics, and every field of science.
Lesson 23 Energy and Eccentricity
According to Newton's laws, all objects in a gravitational field trace out conic sections. The precise shape of an orbit depends on the interplay between energy and eccentricity.
Conic sections have mathematical and grammatical properties:
ellipse - ellipses
parabola - parable
hyperbola - hyperbole
Asteroids follow these conic sections and display how nature obeys mathematics, particularly in the universe.
Newton interpreted Kepler's three laws and developed the equation that models the Earth's orbit:
L^2/DM/(1+ecos(theta)) = r (closely related to the equation of an ellipse)
D = G M M(o)
One of the underlying mathematical properties of the universe is that planets move in ellipses determined by angular momentum, mass of a planet, the mass of the sun, and the eccentricity of the orbit. Not every orbit must be elliptic, circular etc.
The exact shape of an orbit is determined by energy. For example, when a planet falls close to the sun, the potential energy is low, but the planet speeds up so kinetic energy is high. Potential energy is high when a planet is far away from the sun, and kinetic energy is low. Since space is a vacuum, total energy does not change and this is the key to the shape of the orbit.
All the orbits of the planets in the universe are paths of constant energy. Pluto (discovered by Percival Lowell) is responsible for irregularities in Neptune and Uranus's orbit, and its orbit intersects that of Neptune. Halley's comet also has an orbit that is extremely eccentric and elliptical.
The total energy of the orbit is the sum of kinetic and potential energy.
Potential Energy:
U = -D/r
r = L^2/DM/(1 + ecos(theta))
U = D^2M/L^2 (-1-ecos(theta))
Kinetic Energy:
K = (1/2) m v ^2
K = MD^2/L^2((1/2) + ecos(theta) + (1/2)e^2)
E = U+K is constnt because the (cosine) terms cancel in the addition of potential energy and kinetic energy.
E = D^2M/L^2 (1/2) (e^2-1) - connection between energy and eccentricity.
The shapes of orbits depend on energy. For a given angular momentum, the energy determines the eccentricity, and the eccentricity determines the orbit.
Ellipse - if the total energy is negative, and the eccentricity is less than 1, then the orbit is an ellipse. Postive kinetic energy is too small to overcome negative potential energy so the body cannot escape a solar system.
Parabola - if the total energy is zero, and the eccentrcity is exactly 1, then the orbit is a parabola. If a body started with 0 kinetic energy and fell from infinity it would whiponce around the sun and return to infinity. This is very unlikely, however possible.
Hyperbola - if the total energy is positive and the eccentricity is greater than 1, then the orbit is a hyperbola. If an object was projected towards the sun from a great distance, the positive kinetic energy would overcome the negative potential energy. Some comets have this orbit.
Circle - if the total energy has a special value, the eccentricity 0 and the orvit is a circle. This is the lowest energy a planet can have for a given angular momentum. The rings of Saturn display this.
Conic sections have mathematical and grammatical properties:
ellipse - ellipses
parabola - parable
hyperbola - hyperbole
Asteroids follow these conic sections and display how nature obeys mathematics, particularly in the universe.
Newton interpreted Kepler's three laws and developed the equation that models the Earth's orbit:
L^2/DM/(1+ecos(theta)) = r (closely related to the equation of an ellipse)
D = G M M(o)
One of the underlying mathematical properties of the universe is that planets move in ellipses determined by angular momentum, mass of a planet, the mass of the sun, and the eccentricity of the orbit. Not every orbit must be elliptic, circular etc.
The exact shape of an orbit is determined by energy. For example, when a planet falls close to the sun, the potential energy is low, but the planet speeds up so kinetic energy is high. Potential energy is high when a planet is far away from the sun, and kinetic energy is low. Since space is a vacuum, total energy does not change and this is the key to the shape of the orbit.
All the orbits of the planets in the universe are paths of constant energy. Pluto (discovered by Percival Lowell) is responsible for irregularities in Neptune and Uranus's orbit, and its orbit intersects that of Neptune. Halley's comet also has an orbit that is extremely eccentric and elliptical.
The total energy of the orbit is the sum of kinetic and potential energy.
Potential Energy:
U = -D/r
r = L^2/DM/(1 + ecos(theta))
U = D^2M/L^2 (-1-ecos(theta))
Kinetic Energy:
K = (1/2) m v ^2
K = MD^2/L^2((1/2) + ecos(theta) + (1/2)e^2)
E = U+K is constnt because the (cosine) terms cancel in the addition of potential energy and kinetic energy.
E = D^2M/L^2 (1/2) (e^2-1) - connection between energy and eccentricity.
The shapes of orbits depend on energy. For a given angular momentum, the energy determines the eccentricity, and the eccentricity determines the orbit.
Ellipse - if the total energy is negative, and the eccentricity is less than 1, then the orbit is an ellipse. Postive kinetic energy is too small to overcome negative potential energy so the body cannot escape a solar system.
Parabola - if the total energy is zero, and the eccentrcity is exactly 1, then the orbit is a parabola. If a body started with 0 kinetic energy and fell from infinity it would whiponce around the sun and return to infinity. This is very unlikely, however possible.
Hyperbola - if the total energy is positive and the eccentricity is greater than 1, then the orbit is a hyperbola. If an object was projected towards the sun from a great distance, the positive kinetic energy would overcome the negative potential energy. Some comets have this orbit.
Circle - if the total energy has a special value, the eccentricity 0 and the orvit is a circle. This is the lowest energy a planet can have for a given angular momentum. The rings of Saturn display this.
Lesson 22 The Kepler Problem
Kepler's three laws describe the motion of the planets, but Isaac Newton's explanation of these laws was the culmination of the Scientific Revolution.
Isaac Newton hypothesized that the force of gravity gave rise to elliptical orbits:
F = -D/r^2(r^) - inverse square law of gravity
F = m a
m a = -D/r^2(r^)
a = -D/Mr^2 (r^)
d^2r/dt^2 = -D/Mr^2 (r^) - differential equation of any orbit because the solution is the algebraic equation of any conic section.
r = L^2/DM/ 1+ecos(theta)
Twist or torque cannot be applied by gravity rxF =0
F= m a
r x m a = 0
m r x a = 0
m r x dv/dt = 0
d(r x v) = dr/dt x v + r x dv/dt
dt
=v x v + r x dv/dt
d(r x v) = r x dv/dt
dt
m d(r x v) = 0
dt
Zero torque, together with Newton's second law leads to a differential equation to be integrated.
integral(d(mr x v)) = 0
dt
mr x v = L
When there is no torque a certain quantity is constant called angular momentum.
A planet orbiting with constant angular momentum stays in a single plane with orbital speed varying in a precisely determined way. The area swept out by its radius vector changes at a constant rate (Kepler's second law).
da/dt = (1/2) r x v
This is easily determined in polar coordinates where da/dt = 1/2r^2 d(theta)/dt k^ including planets moving in an ellipse under Newton's Universal Law of Gravity.
F = -G M M(o)/r^2 (r^)
= -D/r^2 (r^)
F = m a
a = -D/Mr^2 (r^)
>Cross these vectors
L = Mr^2 d(theta)/dt (k^)
a x L = -D/Mr^2 (r^) x Mr^2 d(theta)/dt (k^)
d(v x L) = D dr^
dt dt
v x L = D (r^ + e)
The orbit of heavens described by the perfect circle of underlying geometry. Product of these events:
r . v x L = Dr1 (r^ + e)
r x v . L = D r . (r^ + e)
Once order is exchanged, and mass allowed its play
L^2/M = Dr . (r^ + e)
L^2/DM = r . (r^ + e)
L^2/DM = (r . r^ + r . e)
= (r + recos(theta))
r = L^2/DM -conic section algebraic equation.
1 + ecos(theta)
The force of gravity moves all heavenly bodies along conic sections.
Isaac Newton hypothesized that the force of gravity gave rise to elliptical orbits:
F = -D/r^2(r^) - inverse square law of gravity
F = m a
m a = -D/r^2(r^)
a = -D/Mr^2 (r^)
d^2r/dt^2 = -D/Mr^2 (r^) - differential equation of any orbit because the solution is the algebraic equation of any conic section.
r = L^2/DM/ 1+ecos(theta)
F= m a
r x m a = 0
m r x a = 0
m r x dv/dt = 0
d(r x v) = dr/dt x v + r x dv/dt
dt
=v x v + r x dv/dt
d(r x v) = r x dv/dt
dt
m d(r x v) = 0
dt
Zero torque, together with Newton's second law leads to a differential equation to be integrated.
integral(d(mr x v)) = 0
dt
mr x v = L
When there is no torque a certain quantity is constant called angular momentum.
A planet orbiting with constant angular momentum stays in a single plane with orbital speed varying in a precisely determined way. The area swept out by its radius vector changes at a constant rate (Kepler's second law).
da/dt = (1/2) r x v
This is easily determined in polar coordinates where da/dt = 1/2r^2 d(theta)/dt k^ including planets moving in an ellipse under Newton's Universal Law of Gravity.
F = -G M M(o)/r^2 (r^)
= -D/r^2 (r^)
F = m a
a = -D/Mr^2 (r^)
>Cross these vectors
L = Mr^2 d(theta)/dt (k^)
a x L = -D/Mr^2 (r^) x Mr^2 d(theta)/dt (k^)
d(v x L) = D dr^
dt dt
v x L = D (r^ + e)
The orbit of heavens described by the perfect circle of underlying geometry. Product of these events:
r . v x L = Dr1 (r^ + e)
r x v . L = D r . (r^ + e)
Once order is exchanged, and mass allowed its play
L^2/M = Dr . (r^ + e)
L^2/DM = r . (r^ + e)
L^2/DM = (r . r^ + r . e)
= (r + recos(theta))
r = L^2/DM -conic section algebraic equation.
1 + ecos(theta)
The force of gravity moves all heavenly bodies along conic sections.
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