Question

Find the magnetic induction of the field at the point O of a loop with current I, whose shape is illustrated in the figure.

• A. $\frac{{\mu }_{0}I}{3\pi }(\frac{3\pi }{2a}+\frac{\sqrt{2}}{b})$
• B. $\frac{{\mu }_{0}I}{2\pi }(\frac{3\pi }{4a}+\frac{\sqrt{2}}{2b})$
• C. $\frac{{\mu }_{0}I}{2\pi }(\frac{2\pi }{3a}+\frac{\sqrt{2}}{b})$
• D. $\frac{{\mu }_{0}I}{2\pi }(\frac{3\pi }{4a}+\frac{\sqrt{2}}{2b})$

Answer 1

The correct option is B

$\frac{{\mu }_{0}I}{2\pi }(\frac{3\pi }{4a}+\frac{\sqrt{2}}{2b})$

$B_C=\frac{{\mu }_{0}I}{4\pi a}\left(\frac{3}{2}\pi \right)$
$B_S=2\times \frac{{\mu }_{0}I}{4\pi b}(sin{0}^{\circ }+sin{45}^{\circ })=\sqrt{2}\times \frac{{\mu }_{0}I}{4\pi b}$
Now, $B=B_C+B_S$
$=\frac{{\mu }_{0}I}{2\pi }(\frac{3\pi }{4a}+\frac{\sqrt{2}}{2b})$