Question

A coil of $n$ turns and radius $r$ is placed in a magnetic field. The electrical power is dissipated due to the current induced in the coil. By stretching the wire, if the number of turns were to be quadrupled and the wire radius halved, the electrical power dissipated would be :

• A. halved
• B. the same
• C. doubled
• D. quadrupled

Answer 1

The correct option is B the same

Induced emf $\left|e\right|=N\frac{d\varphi }{dt}$

Thus, $\left|e\right|\propto N$ ... (i)

(N : number of turns in the coil)

Power dissipated in the coil $P=\frac{e^2}{R}$ ...(ii)

where $R$ is the resistance of the coil.

Now if the wire is stretched, its volume remains unchanged,

$\therefore Al=constant$

$\pi r^{2}l=constant$

or $l\propto \frac{1}{r^2}$ ... (iii)

From eqs. (i) and (ii), Power dissipated in the coil is

$P\propto \frac{N^2}{R}$

or $P\propto N^2\frac{r^2}{l}\propto N^2{r}^4$ [from eq. (iii)]

$\because \frac{P_2}{P_1}={\left(\frac{N_2}{N_1}\right)}^2{\left(\frac{r_2}{r_1}\right)}^4$

radius of wire is halved
$r\to \frac{r}{2}$
no. of turns. $N\to 4N$
$\frac{P_2}{P_1}=1$