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.
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