# Question about Purcell - Electricity and Magnetism

1. Dec 1, 2015

### Diaz Lilahk

In chapter 5 of Purcell's Electricity and Magnetism, he shows using principles of special relatives that the magnetic force of a current carrying wire on a moving charge is equivalent to the charge experiencing an electric force in its own frame of reference.

His argument is that when the current is flowing in the wire, the wire is electrically neutral which means that the moving electrons in the wire are distributed in such a way that along any segment of wire there are equal numbers of electrons as there are holes. So in the lab frame, the charge moving parallel to the direction of the electrons in the wire would experience a magnetic force directed toward or away from the wire depending on the sign or the moving charge. He continues that in the reference frame of the charge moving parallel to the wire which we can assume is greater than the speed of the electrons in the wire. As a result in the reference frame of the charge moving parallel to the wire the distribution of holes undergoes a greater Lorentz length contraction than the electrons in the wire and which would give any segment of wire a net positive charge, therefore the charge would experience an Electric Force.

Now the problem I have with this is that if I turned off the current, this would imply that a stationary charge above the wire would experience an Electric force. Since the distribution of electrons in the wire would undergo a length expansion since they are no longer moving relative to the lab frame, as a result the wire should have a net positive charge and the stationary charge above the wire would experience an Electric force. But this is definitely not what we observe.

What am I missing?

Thank you.

2. Dec 1, 2015

### Ibix

If you turn off the current the electrons in the wire will decelerate to rest with respect to the wire. What will happen to the different Lorentz contractions and the net charge then?

If the answer seems to violate the conservation of charge, you've probably forgotten to take the "return" wire into account. Current flows round a loop.

Does that help?

3. Dec 1, 2015

### A.T.

Length contraction describes the relation between different frames for the same situation, not between different situations (current off / on).

See this post by DrGreg with a great diagram:
https://www.physicsforums.com/showthread.php?p=4528480

4. Dec 2, 2015

### Diaz Lilahk

Thank you, I think this is definitely it.

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