Question

A coil $\text{A}$ of radius $R$ and number of turns $n$ carries a current $i$ and it is placed in a horizontal plane. A small conducting ring $\text{P}$ of radius $r(r<<R)$ is placed at a height $y$ above the centre of the coil $\text{A}$ as shown in the figure. Calculate the induced $\text{emf}$ in the ring when the ring is allowed to fall freely. Express the induced $\text{emf}$ as a function of instantaneous speed $v$ of the falling ring and its height $y$ above the centre of the coil $\text{A}.$

• A. $\frac{3}{2}\frac{{\mu }_0\pi ni{R}^2{r}^{2}yv}{(R^2+y^2{)}^{3/2}}$
• B. $\frac{3}{2}\frac{{\mu }_0\pi ni{R}^2{r}^{2}yv}{(R^2+y^2{)}^{5/2}}$
• C. $\frac{1}{2}\frac{{\mu }_0\pi ni{R}^2{r}^{2}yv}{(R^2+y^2{)}^{5/2}}$
• D. $\frac{2{\mu }_0\pi ni{R}^2{r}^{2}yv}{(R^2+y^2{)}^{5/2}}$

Answer 1

The correct option is B $\frac{3}{2}\frac{{\mu }_0\pi ni{R}^2{r}^{2}yv}{(R^2+y^2{)}^{5/2}}$

The magnetic induction at a point on the axis of a current carrying coil at a distance $y$ from its centre is given by,

$B=\frac{{\mu }_{0}ni{R}^2}{2(R^2+y^2{)}^{3/2}}$

Here,
$R=$ Radius of coil
$n=$ number of turns

Due to the coil $\text{A}$ the magnetic flux linked with ring $\text{P}$ is given as,

$\varphi =BA\cos 0^{\circ }$

$\varphi =\frac{{\mu }_{0}ni{R}^2}{2(R^2+y^2{)}^{3/2}}\times \pi r^2$

$\varphi =\frac{{\mu }_0\pi ni{R}^2{r}^2}{2(R^2+y^2{)}^{3/2}}$

If the ring $\text{P}$ falls with instantaneous velocity $v$ at any instant of time, the magnitude of induced $\text{emf}$ induced in the ring $\text{P}$ is given as,

$E=-\frac{d\varphi }{dt}=-\frac{{\mu }_0\pi ni{R}^2{r}^2}{2}\times \frac{d}{dt}\left[\right(R^2+y^2{)}^{-3/2}]$

$=-(-\frac{3}{4}{\mu }_0\pi ni{R}^2{r}^2(R^2+y^2{)}^{-5/2}\times 2y\frac{dy}{dt})$

$=\frac{3}{4}\frac{{\mu }_0\pi ni{R}^2{r}^2}{(R^2+y^2{)}^{5/2}}\left(2yv\right)[\because v=\frac{dy}{dt}]$

$\therefore E=\frac{3}{2}\frac{{\mu }_0\pi ni{R}^2{r}^{2}yv}{(R^2+y^2{)}^{5/2}}$

Hence, option $\left(B\right)$ is correct.