Question

Calculate magnetic induction at point O if the wire carrying a current I has the shape shown in figure. The radius of the curved part of the wire is equal to R and linear parts of the wire are very long.

Answer 1

Circuit segment shown in can be considered in three parts
I. A circular loop in y-z plane. since this loop is made of uniform wire, therefore magnetic induction at O due to it is $B_1=0$
ii. A straight part, parallel to x-axis .magnetic induction due to it is ,
$B_2=\frac{{\mu }_{0}I}{4\pi R}(-\hat{k})=-\frac{{\mu }_{0}I}{4\pi R}\hat{k}$
iii. A straight pat in y-z plane . perpendicular distance of O from is of this straight part is r = R $cos{45}^0$ as shown in fig.
Angles subtended by lines joining O and ends of this straight part with perpendicular drawn from O are $\alpha =-{45}^0$ and $\beta ={90}^0$
Magnetic induction at O due to this part is
$B_3=\frac{{\mu }_{0}I}{4\pi R}(sin\alpha +sin\beta )$
$\overrightarrow{B_3}=\frac{{\mu }_{0}I}{4\pi R}(\sqrt{2}-1)\hat{i}$
$\overrightarrow{B}$ =$\overrightarrow{B_1}+\overrightarrow{B_2}+\overrightarrow{B_3}$ = $-\frac{{\mu }_{0}I}{4R}(\sqrt{2}-1)\hat{i}-\frac{{\mu }_{0}I}{8\pi R}\hat{k}$