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- TL;DR Summary
- Understanding how both magnetic and electric fields change in a moving frame
I want to understand how electric and magnetic fields change as measured from an inertial frame ##S## vs. as measured from an inertial frame ##\bar {S}## (which has uniform speed v wrt ##S##). I am working out the following example to do so:
We have a cylindrical symmetric wire of radius R, with constant charge density ##\rho## and current density ##j##. (i.e the current I is uniformly distributed over the wire of circular cross section; assume the flow goes from left to right).
We know that its magnetic field ##\vec B## is ##\vec B = \frac{\mu I}{2\pi s} \hat {z}## (where ##s## is the radius of the Amperian loop) outside the wire while ##\vec B = 0## inside the wire. Let's draw out attention to the external magnetic field.
We know that its electric field ##\vec E## is ##\vec E = E \hat {r}## (where ##\hat {r}## accounts for radial direction).
Now let the wire be in an inertial frame ##\bar {S}##, which has uniform speed v (from left to right) wrt your frame ##S##.
a) Could the electric field become ##\vec E = 0## as measured in frame ##\bar {S}##? Why?
b) Could the magnetic field become ##\vec B = 0## as measured in frame ##\bar {S}##? Why?
I think that time dilation plays no role on changing neither ##\vec E## or ##\vec B## because these are uniform (time-independent) fields. Thus, let's focus on Lorentz contraction (moving objects get shortened).
a) ##E## won't change because Lorentz contraction does not apply. ##E## is perpendicular to the velocity of the frame; dimensions perpendicular to the velocity are not contracted.
b) The magnetic field should be contacted, so it changes. But does that mean that we could make it become zero?
Thanks
We have a cylindrical symmetric wire of radius R, with constant charge density ##\rho## and current density ##j##. (i.e the current I is uniformly distributed over the wire of circular cross section; assume the flow goes from left to right).
We know that its magnetic field ##\vec B## is ##\vec B = \frac{\mu I}{2\pi s} \hat {z}## (where ##s## is the radius of the Amperian loop) outside the wire while ##\vec B = 0## inside the wire. Let's draw out attention to the external magnetic field.
We know that its electric field ##\vec E## is ##\vec E = E \hat {r}## (where ##\hat {r}## accounts for radial direction).
Now let the wire be in an inertial frame ##\bar {S}##, which has uniform speed v (from left to right) wrt your frame ##S##.
a) Could the electric field become ##\vec E = 0## as measured in frame ##\bar {S}##? Why?
b) Could the magnetic field become ##\vec B = 0## as measured in frame ##\bar {S}##? Why?
I think that time dilation plays no role on changing neither ##\vec E## or ##\vec B## because these are uniform (time-independent) fields. Thus, let's focus on Lorentz contraction (moving objects get shortened).
a) ##E## won't change because Lorentz contraction does not apply. ##E## is perpendicular to the velocity of the frame; dimensions perpendicular to the velocity are not contracted.
b) The magnetic field should be contacted, so it changes. But does that mean that we could make it become zero?
Thanks