Question

The magnetic induction at the point O in the adjacent figure is

• A. $\frac{{\mu }_{0}i}{8}(\frac{3}{a}+\frac{1}{b})$
• B. $\frac{{\mu }_{0}i}{4}(\frac{3}{a}+\frac{1}{b})$
• C. $\frac{{\mu }_{0}i}{8\pi }(\frac{3}{a}+\frac{1}{b})$
• D. None of these

Answer 1

The correct option is A $\frac{{\mu }_{0}i}{8}(\frac{3}{a}+\frac{1}{b})$

We know,

Magnetic field due to an arc,

$B=\frac{{\mu }_{0}I}{4R}\times \theta$

For arc of radius a, $\theta =\frac{3\pi }{2}$

$B_a=\frac{{\mu }_{0}I}{4\pi a}\times \frac{3\pi }{2}=\frac{3{\mu }_{0}I}{8a}$

For arc of radius b, $\theta =\frac{\pi }{2}$

$B_b=\frac{{\mu }_{0}I}{4\pi b}\times \frac{\pi }{2}=\frac{{\mu }_{0}I}{8b}$

As both the fielda are in same direction,(into the plane),

$B=B_a+B_b=\frac{{\mu }_{0}I}{8}(\frac{3}{a}+\frac{1}{b})$

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