Question

For the figure given below, the total magnetic induction at point O (due to curved and straight portion) will be

• A. $\frac{{\mu }_{0}i}{2\pi r}$
• B. $\frac{{\mu }_{0}i}{2\pi r}(\pi -\varphi +tan\varphi )$
• C. $\frac{{\mu }_{0}i}{\pi r}(\pi -\varphi +tan\varphi )$
• D. 0

Answer 1

The correct option is B

$\frac{{\mu }_{0}i}{2\pi r}(\pi -\varphi +tan\varphi )$

Distance of the wire from point O, $a=rcos\varphi$
Now, $B_{net}=B_1+B_2=\frac{{\mu }_{0}i}{4\pi r}\theta +\frac{{\mu }_{0}i}{4\pi a}(sin\varphi +sin\varphi )$
$=\frac{{\mu }_{0}i}{4\pi r}(2\pi -2\varphi )+\frac{{\mu }_{0}i}{4\pi rcos\varphi }\left(2sin\varphi \right)$
$=\frac{{\mu }_{0}i}{2\pi r}(\pi -\varphi +tan\varphi )$