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VortexLattice
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Hi, I'm reading the section about "How the fields transform" in Griffiths' Intro to Electrodynamics book, but I'm a little confused and want to understand this fully.
He starts off with the example of a parallel plate capacitor where each plate has charge density \sigma _0. So in the rest frame S_0 of the plates, the electric field in between the plates is E_0 = \sigma _0 /\epsilon_0. Then he says, let's look at a frame (S) that's moving parallel to the plates (like, along them) at speed v_0. Then, because the length of the plates is contracted in this frame but the charge is constant, the charge density increases in this frame and thus the E field increases to E = \gamma_0 E_0.
So this makes sense. Then he says he wants to look at magnetic fields. So he says, in frame S, there's a magnetic field due to the surface currents on each plate: K = \sigma v_0. I assume these "surface currents" are from the fact that in frame S, it look like the charge on the plates is moving at the speed S is moving with respect to S_0. If this is true, that makes sense too.
Then he introduces yet another frame, \overline{S}, that's moving at speed v relative to speed S. He derives the equations for the fields in this frame in terms of the fields in frame S. But this is what I'm confused about: He looks at two "special cases", when E = 0 and when B = 0, in frame S. But how can either equal zero? I guess B could be 0 if frame S isn't actually moving at all with respect to S_0, so it's basically the same as S, but I still can't see how E would be 0. I thought we basically chose to look at S because it necessarily has a nonzero magnetic field.
If anyone has a copy handy, it's on pages 525-532. Can anyone help me?
Thanks!
He starts off with the example of a parallel plate capacitor where each plate has charge density \sigma _0. So in the rest frame S_0 of the plates, the electric field in between the plates is E_0 = \sigma _0 /\epsilon_0. Then he says, let's look at a frame (S) that's moving parallel to the plates (like, along them) at speed v_0. Then, because the length of the plates is contracted in this frame but the charge is constant, the charge density increases in this frame and thus the E field increases to E = \gamma_0 E_0.
So this makes sense. Then he says he wants to look at magnetic fields. So he says, in frame S, there's a magnetic field due to the surface currents on each plate: K = \sigma v_0. I assume these "surface currents" are from the fact that in frame S, it look like the charge on the plates is moving at the speed S is moving with respect to S_0. If this is true, that makes sense too.
Then he introduces yet another frame, \overline{S}, that's moving at speed v relative to speed S. He derives the equations for the fields in this frame in terms of the fields in frame S. But this is what I'm confused about: He looks at two "special cases", when E = 0 and when B = 0, in frame S. But how can either equal zero? I guess B could be 0 if frame S isn't actually moving at all with respect to S_0, so it's basically the same as S, but I still can't see how E would be 0. I thought we basically chose to look at S because it necessarily has a nonzero magnetic field.
If anyone has a copy handy, it's on pages 525-532. Can anyone help me?
Thanks!