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

The correct option is D $\frac{{\mu }_{0}I}{3\pi R}$ at point $B$

Current density: $\rho =\frac{1}{\pi R^2-\pi {\left(R/2\right)}^2}\Rightarrow \rho =\frac{4I}{3\pi R^2}$

Current in smaller cylinder (if there were): $I_1=\rho \pi {\left(\frac{R}{2}\right)}^2=\frac{1}{3}$

For $A:B_A=B_{whole-cylinder}-B_{small-cylinder}$

$\Rightarrow B_A=0-\frac{{\mu }_0\left(I/3\right)}{2\pi \left(R/2\right)}=\frac{{-\mu }_{0}I}{3\pi R}$

For $B:B_B=B_{whole-cylinder}-B_{small-cylinder}$

$=\frac{{\mu }_0(I+I/3)\left(R/2\right)}{2\pi R^2}-0=\frac{{\mu }_{0}I}{3\pi R}$