Question

A rectangular wire loop of sides a and b is placed in a non – uniform magnetic field which varies with x as B = kx where k is a constant. The magnetic field is directed perpendicular to the plane of the coil as shown in fig. The magnetic flux through the coil is

• A. zero
• B. $Ka{b}^2$
• C. $\frac{1}{2}ka{b}^2$
• D. $\sqrt{2}ka{b}^2$

Answer 1

The correct option is C

$\frac{1}{2}ka{b}^2$

Since the magnetic field varies with x, we find the flux by considering a small element of the loop of width dx and length a at a distance x from O, as shown in Fig.

The Total magnetic flux is

$\varphi =\int BdA={\int }_0^{b}kx\left(adx\right)$

$=ka{\int }_0^{b}xdx=\frac{1}{2}ka{b}^2$