Question

From a cylinder of radius $R$, a cylinder of radius $R / 2$ is removed, as shown in figure. Current flowing in the remaining cylinder is $I$. Then, magnetic field strength is

• A. zero at point $A$
• B. zero at point $B$
• C. $\frac{\mu_{0} I}{2 \pi R}$ at point $A$
• D. $\frac{\mu_{0} I}{3 \pi R}$ at point $B$

Answer 1

Answer: (D)

Current density: $\rho=\frac{I}{\pi R^{2}-\pi(R / 2)^{2}}$
$ \Rightarrow \rho=\frac{4 I}{3 \pi R^{2}}$
Current in smaller cylinder (if there were): $I_{1}=\rho \pi\left(\frac{R}{2}\right)^{2}=\frac{I}{3}$
For $A: B_{A}=B_{\text {whole-cylinder }}-B_{\text {small-cylinder }}$
$\Rightarrow B_{A}=0-\frac{\mu_{0}(I / 3)}{2 \pi(R / 2)}=\frac{-\mu_{0} I}{3 \pi R}$
For $B: B_{B}=B_{\text {whole-cylinder }}-B_{\text {small-cylinder }}$
$=\frac{\mu_{0}(I+I / 3)(R / 2)}{2 \pi R^{2}}-0=\frac{\mu_{0} I}{3 \pi R}$