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: $\rho =\frac{I}{\pi R^2-\pi (R/2{)}^2}$
$\Rightarrow \rho =\frac{4I}{3\pi R^2}$
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 $B:B_B=B_{\text{whole-cylinder }}-B_{\text{small-cylinder }}$
$=\frac{{\mu }_0(I+I/3)(R/2)}{2\pi R^2}-0=\frac{{\mu }_{0}I}{3\pi R}$