Question

At what value of x is the magnitude of $\overrightarrow{B}$ is maximum ?

• A. $x=0$
• B. $x=\sqrt{2}a$
• C. $x=\pm a/2$
• D. $x=\pm a$

Answer 1

The correct option is D $x=\pm a$

At a position on the x-axis :
$B_{net}=2\frac{{\mu }_{0}I}{2\pi r}cos\theta =\frac{{\mu }_{0}I}{\pi \sqrt{x^2+a^2}}\frac{x}{\sqrt{x^2+a^2}}$
$B_{net}=\frac{{\mu }_{0}Ix}{\pi (x^2+a^2)}$
its graph is shown below.
The magnetic field is maximum when :
$\frac{dB}{dx}=0=\frac{C}{x^2+a^2}-\frac{2C{x}^2}{(x^2+a^2{)}^2}$
$(x^2+a^2)-2x^2\Rightarrow x=\pm a$ .