How is an electric field induced in a wire loop moving through a B field?

[cragar]

Im reading a page out of Griffiths about 2 guys holding a wire loop on a railroad car and then traveling through a B field on the track. As the guys travel at a speed v through the B field the electrons will start to move in the conductor. If we view it from the point of the the B field that is created by 2 sheet currents, and these metal plates that create the sheet currents are at rest relative to the rail road. So if I am standing next to these plates and I see these 2 guys go by at a speed v through the B field. I would say that the electrons in the wire moved because of the Lorentz force. Now Griffiths says that if we view it from the railroad car that we will say the electrons moved because of an electric field. But if we are on the rail road car we would see the B field coming towards us. And he says that a changing B field induces an E field. So by Faradays law we will get an emf in the loop. But it seems strange to think of it as an induced E field. Because it is not like we are changing the current. It seems better to think of it in terms of length contraction. The density of the electrons will be length contracted differently then the protons so I will see a net E field because now I have free charge. Is this what he means by induced E field?

 

[G01]

In that section Griffith's is pointing out a convienent coincidence in the flux law and faraday's law, which he teaches as being separate pehnomena. (See the corresponding sections in chapter 7 of Griffiths.)

In the ground frame, the magnet is not moving and the currents don't change, so the B field is constant. However, the wire loop moves and thus the B field flux through the loop changes. Thus, by the flux rule, currents are induced in the loop.

In the train car frame, the wire loop is at rest. The currents in the magnet have not changed, but the magnet is now moving. so the field moves with it. Thus, the B-Field is not constant in this frame, even though we haven't changed the current. Thus, by Faraday's, the exact same current as in the ground frame is induced by the changing B-Field by the flux law.

Griffith's is using this example to illustrate how Maxwell's electrodynamics can be valid in all inertial frames, despite the fact that it may not seem so and first glance.

 

[cragar]

ok that makes sense, thanks for the answer.
If we view it from the railroad car frame the wire loop is at rest, But because we observe a current flowing in the loop there must have been an E field to cause this because B fields can't do work and move charges from rest. Is this how we know there must have been an E field in this frame.

 

[G01]

ok that makes sense, thanks for the answer.
If we view it from the railroad car frame the wire loop is at rest, But because we observe a current flowing in the loop there must have been an E field to cause this because B fields can't do work and move charges from rest. Is this how we know there must have been an E field in this frame.

Yes, B does not do the work. The induced E field does.

 

[cragar]

Ok and the fact that their was an E field in one frame and no E field in other frame led Einstein to further investigate it.

 

[Phrak]

Electric and magnetic fields don't move, guys. Fields do not have an associated velovity. If vectors, they have a direction and a magnitude.

 

[G01]

Electric and magnetic fields don't move, guys. Fields do not have an associated velovity. If vectors, they have a direction and a magnitude.

The non zero magnetic field values will move along with the magnet.

The point is that in the trains rest frame, the wire loop sees a changing magnetic field due to the relative motion of the magnet, and thus an electric field is induced in the wire by Faraday's law.

In the ground's frame, the magnetic field is constant in time and space, and the loop moves through the field. The change in flux caused by the motion of the loop sets up a magnetic force in the loop, causing a current to flow.

In one frame, the force is electric in nature. In the other, it's magnetic. However, the end result, the current, is the same. Thus, this simple example is evidence that Maxwell's Electrodynamics obeys the principle of relativity. See Griffith's Electrodynamics p. 477-478

 

[jtbell]

In the ground's frame, the magnetic field is constant in time and space, and the loop moves through the field. The change in flux caused by the motion of the loop sets up a magnetic force in the loop, causing a current to flow.

In this frame, I would say that the magnetic force is simply the Lorentz force \vec F = q \vec v \times \vec B which is caused by the motion of the electrons through the magnetic field. The change in magnetic flux through the moving loop, in this frame, is simply a convenient way to calculate the net effect of the Lorentz force on all the electrons.

 

[G01]

In this frame, I would say that the magnetic force is simply the Lorentz force \vec F = q \vec v \times \vec B which is caused by the motion of the electrons through the magnetic field. The change in magnetic flux through the moving loop, in this frame, is simply a convenient way to calculate the net effect of the Lorentz force on all the electrons.

Yes, this approach is definitely more physically transparent. I was just trying to stick with the flux argument, as that is how Griffith's presents it in the discussed section.

 

[Phrak]

The non zero magnetic field values will move along with the magnet.

Yes, the field values would roughly (v<c, and near fields) comove with the magnet. The fields themselves are defined as bound vectors in any inertial frame.