Find the magnetic induction of the field at the point $O$  if the wire carrying a current $I$  has the shape shown in the given figure $a, b, c.$The radius of the curved part of wire is $R$, the linear parts of the wire are very long. 

The magnetic induction due to straight part is

 ${\stackrel{\rightharpoonup }{B}}_{straight}=2\left[\frac{{\mu }_{0}I}{4\mathrm{\pi R}}\right]\left(-\hat{k}\right),$ and due to curved part is 

${\stackrel{\rightharpoonup }{B}}_{curved}=\left[\frac{{\mu }_0\left({\displaystyle \frac{1}{2}}\right)I}{2R}\right]\left(-\hat{i}\right)$

Thus total magnetic induction

$\stackrel{\rightharpoonup }{B}={\stackrel{\rightharpoonup }{B}}_{straight}+{\stackrel{\rightharpoonup }{B}}_{curved}$

$=2\left[\frac{{\mu }_{0}I}{4\pi R}\right]\left(-\hat{k}\right)+\left[\frac{{\mu }_{0}I}{4R}\right]\left(-\hat{i}\right)$

$=-\frac{{\mu }_{0}I}{4\mathrm{\pi R}}\left[\mathrm{\pi }\hat{\mathrm{i}}+2\hat{\mathrm{k}}\right]$

 $\overrightarrow{B}={\overrightarrow{B}}_{straight}+{\overrightarrow{B}}_{curved}$

$=\left[\frac{{\mu }_{0}I}{4\mathrm{\pi R}}\left(-\hat{k}\right)+\frac{{\mu }_{0}I}{4\mathrm{\pi R}}\left(-\hat{i}\right)\right]+\left[\frac{{\mu }_0\left({\displaystyle \frac{1}{2}}\right)I}{2R}\right]\left(-\hat{i}\right)$

$=\frac{{\mu }_{0}I}{4\mathrm{\pi R}}\left[\left(\mathrm{\pi }+1\right)\hat{i}+\hat{k}\right]$

 In this case current in the curved conductor is divided into two parts. The length of upper curved part is thrice that of hidden lower part and so currents in them are $\frac{I}{4}$ and $\frac{3I}{4}$respectively. Thus

$\stackrel{\rightharpoonup }{B}={\stackrel{\rightharpoonup }{B}}_{straight}+{\stackrel{\rightharpoonup }{B}}_{curved}$

$=\left[\frac{{\mu }_{0}I}{4\mathrm{\pi R}}\left(-\hat{k}\right)+\frac{{\mu }_{0}I}{4\mathrm{\pi R}}\left(-\hat{j}\right)\right]+\left[\frac{{\mu }_0\left({\displaystyle \frac{3}{4}}\right)\left({\displaystyle \frac{I}{4}}\right)}{2R}\left(-\hat{i}\right)+\frac{{\mu }_0\left({\displaystyle \frac{1}{4}}\right)\left({\displaystyle \frac{3I}{4}}\right)}{2R}i\right]$

$=-\frac{{\mu }_{0}I}{4\mathrm{\pi R}}\left(\hat{j}+\hat{k}\right)$