Consider a positively charged particle moving at constant speed parallel to a current-carrying wire, in the direction of the current. As you know (after studying Chapters 27 and 28), the particle is attracted to the wire by the magnetic force due to the current. Now suppose another observer moves along with the particle, so according to him the particle is at rest. Of course, a particle at rest feels no magnetic force. Does that observer see the particle attracted to the wire or not? How can that be? (Either answer seems to lead to a contradiction: If the particle is attracted, it must be by an electric force because there is no magnetic force, but there is no electric field from a neutral wire; if the particle is not attracted, you see that the particle is, in fact, moving toward the wire.)

Step by step solution

01

Reference frame of a stationary observer

From the stationary observer's point of view, the positively charged particle moves parallel to the current-carrying wire. Since the wire carries a current, it creates a magnetic field perpendicular to the current direction. The charged particle, being a moving charge, experiences a magnetic force due to this magnetic field which attracts it towards the wire.

02

Reference frame of a moving observer

Now, consider an observer moving along with the particle. This observer sees the particle at rest, so there seems to be no magnetic force acting on it. However, from this observer's perspective, the electrons in the wire are moving in the opposite direction, with a velocity equal and opposite to that of the positively charged particle.

03

Relativistic effect on electron density

According to the theory of relativity, the moving electrons in the wire appear to be more densely arranged for the moving observer, due to length contraction. It means that charges in the wire seem to be more negatively charged for the moving observer than for the stationary observer. This creates an electric field in the vicinity of the wire.

04

Electric force on the particle

For the moving observer, since the charged particle is at rest, it does not experience any magnetic force. However, due to the electric field generated by the relativistically contracted electron density, the positively charged particle experiences an electric force, which attracts the particle towards the wire.

05

Resolving the contradiction

The seeming contradiction between the two reference frames can be resolved using the theory of relativity. In the stationary observer's frame, the magnetic force acts on the moving particle, attracting it towards the wire. While, in the moving observer's frame, the electric force resulting from the relativistically contracted electron density attracts the particle towards the wire. So, in both frames, the particle is observed to be attracted to the wire, but due to different forces. This resolves the apparent contradiction in the problem.

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