Question

$\text{A filamentary current of 10 A is directed.in from infinity to the origin on the positive x-axis.}$
$\text{The magnetic field intensity}\overrightarrow{H}\text{at (0,0, 1) is}$

• A. $\frac{5}{2\pi }{\hat{a}}_z$
• B. $\frac{10}{4\pi }{\hat{a}}_x$
• C. $\frac{5}{4\pi }{\hat{a}}_x$
• D. $\frac{10}{4\pi }{\hat{a}}_y$

Answer 1

The correct option is D $\frac{10}{4\pi }{\hat{a}}_y$

According to the Biot=Savart’s law,

$\overrightarrow{H}=\int \frac{\overrightarrow{dL}\times \overrightarrow{R}}{4\pi R^3}$

$\overrightarrow{R}=-x{\hat{a}}_x+{\hat{a}}_z$

$R=\sqrt{1+x^2}$

$\overrightarrow{H}={\int }_{\infty }^0\frac{10dx(-{\hat{a}}_x)\times (-x{\hat{a}}_x+{\hat{a}}_z)}{4\pi (x^2+1{)}^{3/2}}$

$\overrightarrow{H}=\frac{10}{4\pi }{\int }_{\infty }^0\frac{dx}{(x^2+1{)}^{3/2}}{\hat{a}}_y$

$\overrightarrow{H}=\frac{10}{4\pi }\frac{x}{\sqrt{1+x^2}}{|}_{\infty }^0{\hat{a}}_y=\frac{10}{4\pi }{\hat{a}}_y$