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.

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.

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.

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))

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.

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.

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.