In the given figure, the magnetic field at $O$.

• A

  $\frac{3}{4} \frac{\mu_{0} I}{r}+\frac{\mu_{0} I}{4 \pi r}$
• B

  $\frac{3}{10} \frac{\mu_{0} I}{r}-\frac{\mu_{0} I}{4 \pi r}$
• Correct answer
  C

  $\frac{3}{8} \frac{\mu_{0} I}{r}+\frac{\mu_{0} I}{4 \pi r}$
• D

  $\frac{3}{8} \frac{\mu_{0} I}{r}-\frac{\mu_{0} I}{4 \pi r}$

Solution

According to the figure, magnetic field at point is given as

$B_{\text {net }}=$ magnetic field due to straight wire $A B+$ magnetic field due to straight wire $C D+$ magnetic field due circular wire $B C$ Since, point $O$ is along the length of the wire $B A$. So, $B_{A B}=0$ $\begin{aligned} B_{\text {net }} &=0+\frac{\mu_{0}}{4 \pi} \cdot \frac{I}{r}+\frac{270}{360}\left(\frac{\mu_{0} I}{2 r}\right) \\ &=\frac{\mu_{0} I}{4 \pi r}+\frac{3}{4} \cdot \frac{\mu_{0} I}{2 r} \\ &=\frac{\mu_{0} I}{4 \pi r}+\frac{3 \mu_{0} I}{8 r}(\text { downward }) \end{aligned}$

Hence, the correct answer is option: 'C'.