Question

A square frame of side carries i produces a field B at its centre. The same current is passed through a circular coil having the same perimeter as the square. The field at the centre of circular coil is B'. Find the ratio of (B'/B).

• A. $\frac{{\pi }^2}{3\sqrt{2}}$
• B. $\frac{{\pi }^2}{5\sqrt{2}}$
• C. $\frac{{\pi }^2}{7\sqrt{2}}$
• D. $\frac{{\pi }^2}{8\sqrt{2}}$

Answer 1

The correct option is D $\frac{{\pi }^2}{8\sqrt{2}}$

$\text{Case: square frame}$

$B=4\times \frac{{\mu }_{0}I}{4\pi \frac{a}{2}}\times (\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}})$

$B=\frac{{\mu }_{0}I2\sqrt{2}}{\pi a}$

$\text{Case: loop}$

$4a=2\pi R$

$R=\frac{2a}{\pi }$

$B^{\prime }=\frac{{\mu }_{0}I}{2\times \frac{2a}{\pi }}$

$\frac{B^{\prime }}{B}=\frac{{\mu }_{0}I\pi }{4a}\times \frac{\pi a}{{\mu }_{0}I2\sqrt{2}}=\frac{{\pi }^2}{8\sqrt{2}}$

Option $\text{D}$ is the correct answer.