Re: Ohm/Kirchoff in special relativity

On Jan 30, 10:14 pm, Dono <sa...@xxxxxxxxxxx> wrote:
Is there any treatment in literature for the relativistic form of Ohm/
Kirchoff equations? I have never seen one.

I'd be flabbergasted if there wasn't a published instance of a
relativistic Ohm's law since the v << c version of Ohm's law I used
all last semester was derived using the relativistic transformation
rules for E and B. On the other hand, I don't expect there to be much
demand for relativistic circuit equations.

Ohm's law j = \sigma E is [macroscopically] true in the rest frame of
some observer. That much can be assumed.

Now transform to a new reference frame moving at some velocity v, and
you measure E'. The transformation equation is simply being cited, you
can find it in any electrodynamics textbook that doesn't ignore
relativity.

E' = 1/\gamma [E + v / |v|^2 * ( v.E )(1-\gamma) + v x B)]

In this case, v is in some arbitrary direction.

Transform j ---> j'.

In the v << c [\gamma ~= 1] case, Ohm's law wrt a moving frame becomes
j' = \sigma E' = \sigma [ E + v x B ].

The thing that I had to think about for a minute is what happens to
\sigma. It is a tensorial quantity, and has to transform as such. But
\sigma is a 3-tensor and I can't simply smack it with a Lorentz
transform and be done with it. I think the proper thing to do would be
to pretend that the conductivity tensor is a sub-part of a four tensor
which picks up a Lorentz transform. In that case, if you think about
the matrix algebra, the conductivity tensor would only pick up
influences from the spatial part of the Lorentz transform.

So if you assume a transformation to an observer moving along the x
axis with velocity v, you would get :

j' = \sigma' * [ 1/\gamma [E + v / |v|^2 * ( v.E )(1-\gamma) + v x
B)] ] with \sigma'_xx = gamma * \sigma_xx, with the rest of the
components unchanged. This matches what I'd expect because it reduces
properly to the nonrelativistic case which I'm very familiar with.

Does this help?
.

Relevant Pages

  • Re: Ohm/Kirchoff in special relativity
    ... relativistic Ohm's law since the v << c version of Ohm's law I used ... relativity. ... and has to transform as such. ... to pretend that the conductivity tensor is a sub-part of a four tensor ...
    (sci.physics.relativity)
  • Re: Ohm/Kirchoff in special relativity
    ... relativistic Ohm's law since the v << c version of Ohm's law I used ... relativity. ... and has to transform as such. ... to pretend that the conductivity tensor is a sub-part of a four tensor ...
    (sci.physics.relativity)
  • Re: Ohm/Kirchoff in special relativity
    ... relativistic Ohm's law since the v << c version of Ohm's law I used ... relativity. ... and has to transform as such. ... to pretend that the conductivity tensor is a sub-part of a four tensor ...
    (sci.physics.relativity)
  • Re: Ohm/Kirchoff in special relativity
    ... relativistic Ohm's law since the v << c version of Ohm's law I used ... relativity. ... and has to transform as such. ... to pretend that the conductivity tensor is a sub-part of a four tensor ...
    (sci.physics.relativity)
  • Re: Einstein (1905) Absurdities
    ... Einstein Absurdities ... matin equations at the heart of Special Relativity. ... The really strange and marvelous magical gamma absurdity. ... This creates a transform formula that could be valid only ...
    (sci.physics.relativity)