Question

A coil of radius R carries a current I. Another concentric coil of radius $r(r<<R)$ caries current $\frac{I}{2}$. Initially planes of the two coils are mutually perpendicular and both the coils are free to rotate about common diameter. They are released from rest from this position. The masses of the coils are M and m respectively $(m<M)$. During the subsequent motion let $K_1$ and $K_2$ be the maximum kinetic energies of the two coils respectively and let U be the magnitude of maximum potential energy of magnetic interaction of the system of the coils. Choose the correct options.

• A. $\frac{K_1}{K_2}=\frac{M}{m}{\left(\frac{R}{r}\right)}^2$
• B. $K_1=\frac{{Umr}^2}{{mr}^2+{MR}^2}{K}_2=\frac{{UMR}^2}{{mr}^2+{MR}^2}$
• C. $U=\frac{{\mu }_0\pi I^2{r}^2}{4R}$
• D. $K_2>>K_1$

Answer 1

Answer: (C)

$U=\frac{{\mu }_0\pi I^2{r}^2}{4R}$

	Coil 1	Coil 2
Radius	R	r(<<R)
Current	I	I/2
Mass	M	m
Kinetic energy	$K_1$	$K_2$

$K.E=\frac{1}{2}I{w}^2$ (1)

$\therefore K-1=\frac{1}{2}\times I{w}^2⟹I\propto \frac{1}{K.E}$

$K_2=\frac{{MR}^2}{2}{w}^2$

Now ${K_1/K}_2=\frac{m}{m}{\left(\frac{r}{R}\right)}^2$ and $K_1+K_2=U$ (ii)

$⟹K_2\times \frac{m}{M}(r/R{)}^2+K_2=U$

$⟹K_2=\frac{UM{R}^2}{m{r}^2+M{R}^2}$

Similarly $K_1=\frac{UM{R}^2}{{mr}^2+{MR}^2}$

And using (ii)

$U=U=\frac{1}{2}\frac{\pi r^2{\mu }_{2}I}{2R}=\frac{{\mu }_2{I}^2\pi r^2}{4R}=U$