A given referential circuit R is positioned at a certain distance from a magnetic dipole

Question

A given referential circuit R is positioned at a certain distance from a magnetic dipole a) In a certain reference frame A, the circuit moves with speed v and the dipole is stationary. Use the Lorentz force to show that there will be an induced current in the circuit. What is the direction of the current? b) Describe what happens to the circuit in a reference frame where the circuit is stationary but the dipole moves with speed v. Will there be current? Will it be the same as item a? (Tip: it is not necessary to use any relativistic transformation between the fields)

Answer 1

type in latex to parse

a) In the reference frame A where the circuit moves with speed \(v\) and the magnetic dipole is stationary, we can use the Lorentz force to show that there will be an induced current in the circuit. The Lorentz force (\(F_L\)) on a moving charge in a magnetic field is given by:

\[F_L = q \cdot (v \times B)\]

Where:

- \(F_L\) is the Lorentz force.

- \(q\) is the charge of the particle.

- \(v\) is the velocity of the particle.

- \(B\) is the magnetic field.

In this case, the circuit contains charges that are moving with a velocity \(v\) with respect to the magnetic dipole. As a result, each charge experiences a Lorentz force due to the magnetic field created by the dipole. The charges in the circuit will experience a force that will push them in a certain direction, inducing a current in the circuit.

The direction of the induced current can be determined using the right-hand rule. If you point your right thumb in the direction of the velocity \(v\) of the charges and your fingers in the direction of the magnetic field created by the dipole, then the palm of your hand will indicate the direction of the induced current. The induced current will flow in a direction that creates a magnetic field opposing the change in the external magnetic field due to the dipole. This is in accordance with Lenz's law, which states that the direction of the induced current is such that it opposes the change in the magnetic flux.

b) In the reference frame where the circuit is stationary but the dipole moves with speed \(v\), the situation is different. In this case, the circuit is not moving relative to the magnetic field created by the dipole, so there is no change in magnetic flux through the circuit. According to Faraday's law of electromagnetic induction, an induced electromotive force (emf) and current are generated in a circuit only when there is a change in magnetic flux through the circuit. Since the circuit is stationary and the magnetic field is moving with the dipole, there is no change in magnetic flux, and therefore, there will be no induced current in the circuit in this reference frame.

So, in summary:

a) In reference frame A where the circuit moves, there will be an induced current, and its direction will be determined by the right-hand rule.

b) In a reference frame where the circuit is stationary but the dipole moves, there will be no induced current.