Question

Define self-inductance of a coil.

(a) Obtain an expression for the energy stored in a solenoid of self-inductance ${}^{\prime }{L}^{\prime }$ when the current though it grows from zero to ${}^{\prime }{I}^{\prime }$.
(b) A square loop MNOP of side 20 cm is placed horizontally in a uniform magnetic field acting vertically downwards as shown in the figure. The loop is pulled with a constant velocity of 20 cm $s^{-1}$ till it goes out of the field.
(i) Depict the direction of the induced current in the loop as it goes out of the field. For how long would the current in the loop persist?
(ii) Plot a graph showing the variation of magnetic flux and induced emf as a function of time.

Answer 1

1) Self-inductance- It is defined as the ration of total flux linked with the coil to the current flowing through the coil. It is denoted as $L$.

$L=\frac{{\varphi }_B}{I}$. . . . . . .(1)

Expression for the energy stored in a solenoid-

The induced emf in the coil is given by

$e=-\frac{d{\varphi }_B}{dt}$. . . . .(2)

From equation (1) and (2), we get

$e=-L\frac{dI}{dt}$. . . . . . . . .(3)

The self induced emf is also called back emf, as it opposes any change in the current. So, work needs to be done against back emf in the establishing current and this work done is stored as magnetic potential energy.

The rate of doing work is,

$\frac{dW}{dt}=|e|I=LI\frac{dI}{dt}$

The total work done,

$W=\int dW={\int }_0^{I}LIdI$

$W=\frac{1}{2}L{I}^2$

b)

(i) The direction of induced current in the loop is clockwise.

Given,

$d=20cm$

$v=20cm/s$

The current will persist till the entire loop comes out of the field,

$t=\frac{d}{v}=\frac{20}{20}=1sec$

(ii) The maximum flux is, $\varphi =Bla$, a=side of the square.

The magnetic flux remains constant inside the magnetic field. This flux will starts dropping once the loop comes out of the magnetic field and it is zero when it comes completely out from the magnetic field as shown in the above figure.

The maximum Induced emf in coil is, $e=-\frac{d\varphi }{dt}=-Blv$

The $e$ remains constant till the entire comes out from the magnetic field, when it comes out then emf drops to zero as shown in the above figure.