Question

Two concentric and coplanar circular coils have radii $a$ and $b(>>a)$ as shown in figure. Resistance of the inner coil is $R$. Current in the outer coil is increased from $0$ to $i$, then find the total charge circulating the inner coil :

• A. $\frac{{\mu }_0\pi a^{2}i}{2Rb}$
  
• B. $\frac{2{\mu }_0\pi a^{2}i}{2Rb}$
  
• C. $\frac{{\mu }_0\pi a^{2}i}{4Rb}$
  
• D. $\frac{{\mu }_0\pi a^{3}i}{4Rb}$
  

Answer 1

The correct option is A $\frac{{\mu }_0\pi a^{2}i}{2Rb}$

Magnetic field in the inside region of outer coil due to current $I$ in the coil$=\frac{{\mu }_{0}I}{2b}$

Thus the flux passing through the coil=$BA=\frac{{\mu }_{0}I}{2b}\times \pi a^2=\frac{{\mu }_{0}I\pi a^2}{2b}$

Hence the emf produced in inner coil$=\frac{d\varphi }{dt}$ $=\frac{{\mu }_0\pi a^2}{2b}\frac{dI}{dt}$

Hence the total charge flowing in the inner coil=${\int }_0^t{i}_{inner}dt$ $=\frac{{\mu }_0\pi a^2}{2Rb}{\int }_0^t\frac{dI}{dt}dt$

$=\frac{{\mu }_0\pi a^{2}i}{2Rb}$