Question

A magnetic field is applied perpendicular to the plane of a flat coil of coppered wire. The time variation of the magnetic flux density is given by $B_0\sin (2\pi t/T)$ as shown graphically in the figure.

At which of the following values of $t$ is the magnitude of the emf induced in the coil is maximum ?

• A. $\frac{T}{8}$
• B. $\frac{T}{4}$
• C. $\frac{3T}{8}$
• D. $\frac{T}{2}$

Answer 1

Answer: (D)

Assume the area of the flat coil of copper wire to be $A$.
Then the flux linking the coil is
$\varphi =BA=B_{0}A\sin \left(\frac{2\pi t}{T}\right)$
Thus, the induced emf in the coil is given by
$e=\frac{d\varphi }{dt}=-\frac{2\pi B_{0}A}{T}\cos \left(\frac{2\pi t}{T}\right)$
Maximum of $E$ occurs when $\cos \frac{2\pi t}{T}=-1$
or$\frac{2\pi t}{T}=\pi ie,t=\frac{T}{2}$
Since $e=-\frac{d\varphi }{dt}=-A\frac{dB}{dt}$ is the maximum
it is obvious from the graph that this occurs at
$t=T/2$