Question

A rod closing the circuit shown in the figure moves along a U-shaped wire at a constant speed v under the action of the force $F$ . The circuit is in a uniform magnetic field perpendicular to the plane. Calculate $F$ if the rate of heat generation in the circuit is $Q$ .

(A) $F = Qv$
(B) $F = \dfrac{Q}{v}$
(C) $F = \dfrac{v}{Q}$
(D) $F = \sqrt {Qv}$

Answer 1

An emf is induced when a magnet is taken near a closed conducting coil. Emf can be expanded as an electromotive force. The induced emf is given by the ratio of change in magnetic flux to the change in time. To answer this question we need to substitute the rate of heat dissipation and the force on a current-carrying conductor placed in a magnetic field and rearrange in order to get the desired answer.

Complete Step By Step Answer:
Given that the rod is moving in the U-shaped wire. The given rod is moving under a constant speed $v$ . This is happening under the action of force $F$ . This force can be calculated using the following method.

The rate of heat dissipation is given by an equation that is given below.
$Q = EI$ …… (1)
Here $Q$ is the heat dissipated, $E$ is the emf induced, and $I$ is the current flowing in the rod.
The induced emf or motion emf is defined as emf induced by the motion of the conductor across the magnetic field. The equation of the motional emf is given by,
$E = Bvl$ …… (2)
Here $B$ is the magnetic field, $v$ is the velocity with which the conductor is moving, $l$ is the length of the conductor.
Substituting equation (2) in (1)
$Q = Bvl \times I$ …… (3)
Now the force on a current-carrying conductor placed in a magnetic field is given by,
$F = IlB$ …… (4)
From the above equation if we rearrange for getting the formula for current $I$ we get,
$I = \dfrac{F}{{lB}}$ …… (5)
Substituting this in equation (3) we get,
$Q = Bvl \times \dfrac{F}{{lB}}$
$\Rightarrow Q = Fv$
$\Rightarrow F = \dfrac{Q}{v}$
Correct Answer: Therefore the correct option is B.

Note:
The emf can be induced by a change in magnetic flux. This can happen due to two reasons. One is that if we place the electric conductor in the presence of a changing magnetic field. The emf generated will depend on either a small change in the area vector or a magnetic field. Therefore this will cause an induced emf. The other one is placing a conductor that is moving constantly in a stable and static magnetic field. This in turn causes a change in the area vector and this will generate EMF.