Question

Axis of a solid cylinder of infinite length and radius $R$ lies along $y$-axis. It carries a uniformly distributed current $‘i^{\prime }$ along $+y$ direction. Magnetic field at a point $(\frac{R}{2},y,\frac{R}{2})$ is

• A. $\frac{{\mu }_{0}I}{4\pi R}(\hat{i}-\hat{k})$
• B. $\frac{{\mu }_{0}I}{2\pi R}(\hat{j}-\hat{k})$
• C. $\frac{{\mu }_{0}I}{4\pi R}\hat{j}$
• D. $\frac{{\mu }_{0}I}{4\pi R}(\hat{i}+\hat{k})$

Answer 1

The correct option is A $\frac{{\mu }_{0}I}{4\pi R}(\hat{i}-\hat{k})$

Position of point in the cylinder can be depicted by following figures




Here $OP=\sqrt{x^2+z^2}=\sqrt{(R/2{)}^2+(R/2{)}^2}=\frac{R}{\sqrt{2}}$

Magnetic field inside a cylinder is given by $B=\frac{{\mu }_{0}ir}{2\pi R^2}$
Here $r=\frac{R}{\sqrt{2}}\Rightarrow B=\frac{{\mu }_{0}i}{2\sqrt{2}\pi R}$

Multiplying by unit vector $\frac{(\hat{i}-\hat{k})}{\sqrt{2}}$ in the direction shown , we get,
$\overrightarrow{B}=\frac{{\mu }_{0}i}{4\pi R}(\hat{i}-\hat{k})$