5. Magnetism

In this section we continue the process of determining the magnetic field of moving charge. Since charge can accumulate, that is, increase the intensity of the electric field by increasing the amount of charge in a region of space, it needs to be considered. In keeping with pure field approach, we derive a direct relation between the electric field and a magnetic field based on the velocity of the source of the electric field. It is considered in this context as the fundamental equation of magneto-statics. Normally it is the current term of Maxwell's equations which is considered the fundamental equation, but then again it is perspective, which does not actually change the experimental results. The derivation here is of some interest by itself, since it succeeds in deriving a well known equation by a unique route.

Eq.(4-7) is not complete. Since the electric field can have divergence, there is a further possible contribution to the changing field based on the accumulation of charge in a region of space. It will be necessary to show the change in electric field that results from the increase or decrease of charge (¶r/¶t)¹. If charge is flowing, but not in a loop, then this change in charge density must occur in some region of space. If charge simply appeared out of nowhere, the symmetrical increase in the electrical field would not create a magnetic field. This is why the symmetrical field does not propagate. The electric field tries to decay uniformly, but the magnetic field created cancels itself out. Therefore, if charge is increasing in one place there has to be another place where charge is decreasing (by the same rate). The changing electric field between these two locations can create a magnetic field. Imagine two charges q₁ and q₂ each surrounded by an arbitrary Gaussian surface S₁ and S₂. We will arrange it so that part of these surfaces coincide and call this mutual surface A.



Fig.2: Gaussian surfaces around two charges with an Ampèrian loop at the intersection.

Using the divergence theorem we can write:

(1)

(2)

Note that the two surfaces in each case completely surround the charge. We subtract these equations and obtain

(3)

We must subtract here to obtain the total electric flux through A, since the flux from q₂ is in the opposite direction through the surface. Taking the time derivative of this equation and substituting from Eq.(1-2) yields

(4)

The change in q₁ has to be equal and opposite to q₂. With this fact and Stokes' theorem we have

(5)

Note that while the surfaces are different, the circuit l is the same for all three surfaces, thus

(6)

Where, like the electric flux, B₁-B₂ is just the magnetic flux along the path. This is an Ampere's law² for displacement current. It gives ¶E/¶t when it is related to the flow of charge. We can apply Stokes' theorem to Eq.(4-7), and add this term which yields

(7)

There are no new properties here. This equation is essentially the same as Eq.(1-2). All we have done is to relate the change in electric field to the various causes from the existence of charge (or charge fields). Thus the first term represents only the change of electric field resulting from propagation. The second term represents the change related to the geometry of the longitudinal component (essentially) of the moving electric field. The third term represents the change related to the geometry of the transverse component of the moving electric field. The fourth term represents the change related to the accumulation of charge in a region of space. Now looking at the magnetostatic case where the first term is zero and seeing that the second term describes a current, we can write

(8)

For situations where the total change in charge of a region is caused by the current (I=dq/dt) such as a charged object in motion, the magnetic field is thus (from Eq.(8) and since Ñ×B=0):

(9)

where v is the absolute velocity³. It should be noted here, that this result is exact and true at all velocities as long as there is no acceleration. It turns out that this is the sole equation required (or responsible) for magnetostatics. Note that if I=0 in Eq.(8), the displacement term gives a magnetic field in the same direction as the cross product term. This last term represents the field of a current (see next section) and thus, the displacement term immediately satisfies the continuity conditions in a quasistatic problem (e.g. a circuit with a capacitor).

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1. One might think that this is the whole effect, in place of the velocity terms, since this gives a change in the electric field. However, the cross product is true even if no charge (divergence) exists at all (as seen by Eq.(4-6)). It might best be described as : The velocity terms represent the change in electric field based on its geometry, while the change we are calculating here is based on its quantity.

2. I will call this the accumulation theorem. The results are surprising. Imagine that a space exists with two regions of charge of any shape at any distance of separation. Imagine that one region can increase its charge, but by whatever amount it increases, the other region has to decrease by the same amount. There is no current, the charge is magically transferred. If you picked a random closed path in the space at any location and measured the magnitude of the magnetic field along the path and multiplied it by differential of length of the path, you would know exactly the rate at which the charge was being transferred.(Of course, this is hypothetical as it would always require a current to transfer the charge.)

3. This is the same relation as in Griffiths¹ 426, which we have derived in a more direct fashion.

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