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
$\phi=B A=B_{0} A \sin \left(\frac{2 \pi t}{T}\right)$
Thus, the induced emf in the coil is given by
$e=\frac{d \phi}{d t}=-\frac{2 \pi B_{0} A}{T} \cos \left(\frac{2 \pi t}{T}\right)$
Maximum of $E$ occurs when $\cos \frac{2 \pi t}{T}=-1$
or$\frac{2 \pi t}{T}=\pi \,\,i e, \,\,\,t=\frac{T}{2}$
Since $e=-\frac{d \phi}{d t}=-A \frac{d B}{d t}$ is the maximum
it is obvious from the graph that this occurs at
$t=T / 2$