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- 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.

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