After all these these years dealing with electromagnetism, I still do
not fully understand a Faraday disk based upon Maxwell's equations. To
be specific let me describe a particular configuration.
1. There is a uniform magnetic field in the vertical direction
extending over the region that will contain all the apparatus.
2. The axle mounting a circular disk is vertical so that the disk is in
the horizontal plane. This axis and disk, both of conducting material is
what spins.
3. To make electrical connection, there is a conducting circular
cylindrical skirt attached the disk's periphery. This skirt dips into a
circular trough containing mercury. A similar but smaller skirt and
trough arrangement is used to connect to the axle shaft. Connections to
measure what happens are made to the trough bodies. These wires are not
moved during the testing.
4. These wires are connected to a variable dc source through a
galvanometer. When running, the source is adjusted so that the current
flow is zero. By using this potentiometric technique, there will be no
current to confuse what is happening. Maxwell's equations will also be
simplified.
Maxwell's equations for this case are:
div B = 0. Always true anyway.
div D = 0. There is no free charge. There is the possible
- exception at the disk edge and the like, but I do
- not think this charge is significant.
curl E = 0. There is no time variation of the magnetic field.
curl H = 0. There is no conducted or displacement current.
When the disk turns, how do you explain the potential difference
measured by the external potentiometer?
Something has to be added to the Maxwell equations. For example, I think
that you have to add an equation representing the definition of the
electric field. Charge is a concept that is not explicitly defined but
which can be measured in terms of forces between charges. Given a charge
of size q, the electrical field is determined by F = q E once the
mechanical force is measured. Once you know and measure charge, you know
what current is.
I think that the other thing that needs to be known is equivalent to
knowing how the electromagnetic tensor in 4-space transforms with
motion. that is, relativistically, when a point moves through a magnetic
field, it sees an electric field introduced by the motion at the point.
This electric field may be called the Lorentz force or something
similar, but it seems to be something that must be added to the
equations I have already listed. This arises simply out of relativity
where the EM field has to be a tensor and has to transform properly with
a Lorentz transformation.
Questions:
1. Can this Lorentz force be derived without invoking relativity or
another law of nature? That is, are Maxwell's equations sufficient?
2. Given Maxwell's equations and F = q E, can the forces between wires
carrying current be derived?
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