Question

The total magnetic induction at point O due to curved portions and straight portion in the following figure, will be

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

Answer 1

The correct option is A $\frac{{\mu }_{0}i}{2\pi r}[\pi -\varphi +tan\varphi ]$

$\begin{array}{c}{B}_{{}^{\prime }}=\frac{{\mu }_{o}i}{4\pi \left(r\cos \varphi \right)}\left[\sin \varphi +\sin \varphi \right]\\ =\frac{{\mu }_{o}i}{2\pi r\cos \varphi }\left[2\sin \varphi \right]\\ =\frac{{\mu }_{o}i\tan \varphi }{2\pi r}\\ B_2=\frac{{\mu }_{o}i\left[2\pi -2\varphi \right]}{4\pi r}\\ =\frac{{\mu }_{o}i\left[\pi -\varphi \right]}{2\pi r}\\ \therefore B=\frac{{\mu }_{o}i}{2\pi r}\left[\pi -\varphi +\tan \varphi \right]\\ Hence,\\ optionAiscorrectanswer.\end{array}$