Question

A loop of wire is at the edge of a region of space containing auniform magnetic field $B_vec$ . The plane of the loop isperpendicular to the magnetic field. Now the loop is pulled out ofthis region in such a way that the area of the coil inside the magnetic field region isdecreasing at the constant rate . That is, $\frac{dA}{dt} = - c$ , with .

Part A
The induced emf in the loop is measuredto be . What is the magnitude of the magnetic field that the loop was in?

Part B
For the case of a square loop of sidelength being pulled out of the magneticfield with constant speed (see the figure), what is the rate of change ofarea $c = -\frac{da}{dt}$ ?

Answer 1

Concepts and reason

The required concept to solve this question is Faraday’s Law.

Fin the external magnetic field by using the expression of induced emf.

Fin the rate of change of area by using induced emf and given constants.

Fundamentals

The induced electromotive force in any closed circuit equal to the rate of change of the magnetic flux enclosed by the circuit.

The induced emf in the loop is given as follows:

Here, is the magnetic flux through the loop, A is the area of the loop, and B is the magnetic field.

Let the loop be moving with velocity v in the magnetic field. The motional emf is given as follows:

Here L is the dimension of the square loop and v is the velocity of the loop.

(A)

Consider a square loop of area A placed in magnetic field in such a way that the magnetic field vector B makes angle with area vector

The expression for induced emf in the loop is,

If all the quantities except area of the loop is kept constant, the induced emf in the loop is,

Rewrite the equation for magnetic field as follows:

Substitute V for , for and -c for as follows:

(B)

The motional emf induced in the loop placed perpendicular to magnetic field is given by following expression:

Here L is the dimension of the loop moving out of the magnetic field with constant speed v.

From the last step, the induced emf in the loop is,

Substitute the value in the expression for motional emf:

Ans: Part A

The magnitude of the loop in which the loop was placed is .

Part B

The rate of the change of area of the loop is .