Question

A flat coil of radius 5 cm with 50 turns has a resistance of 0.16 $\Omega$ and is placed in a magnetic field oriented normal to the plane of the coil. The magnetic field varies at a constant rate in time. By switching the magnetic field on at t = 0 s, the rate of heat dissipated in the coil is found to be $0.49J{s}^{-1}$. Then choose the correct options from the following.

• A. the rate of change of magnetic field is $0.7$ Ts${}^{-1}$
• B. the magnetic flux linking the coil varies linearly with time.
• C. the magnetic flux linking the coil is directly proportional to t${}^2$
• D. the induced current through the coil is about $1.72A$

Answer 1

Answer: (A), (B), (D)

the rate of change of magnetic field is $0.7$ Ts${}^{-1}$
the magnetic flux linking the coil varies linearly with time.
the induced current through the coil is about $1.72A$

The rate of change of magnetic field is $\frac{dB}{dt}T/s$
Emf generated in coil is:

$E=\frac{d\varphi }{dt}$
$E=\frac{\pi (5{)}^2}{100\times 100}\times 50\times \frac{dB}{dt}$
$i=\frac{E}{R}=\left(\frac{d\varphi }{dt}\right)\frac{1}{R}=\frac{12.5\pi }{0.16\times 100}\frac{dB}{dt}$

Rate of heat dissipated:

$P=i^{2}R$
$={\left(\frac{12.5\pi }{16}\right)}^2{\left(\frac{dB}{dt}\right)}^2\left(0.16\right)$
$=0.96{\left(\frac{dB}{dt}\right)}^{2}J/s$
Given : $P=0.49$
$⟹0.49=0.96{\left(\frac{dB}{dt}\right)}^2$
$\frac{dB}{dt}\approx 0.7T/s$
$\frac{dB}{dt}$ is constant, therefore, flux changes linearly with time.

Also, since flux $\varphi =BA$ & $B\propto t$, $\varphi \propto t^2$
$i=\frac{12.5\pi }{16}\times 0.7=1.72A$