Figure shows a straight wire length $l$ carrying current $i$. The magnitude of magnetic field produced by the current at point $\mathrm{P}$ is

• A

  $\frac{\sqrt{2} \mu_{0} \mathrm{i}}{\pi l}$
• B

  $\frac{\mu_{0} \mathrm{i}}{4 \pi \mathrm{l}}$
• Correct answer
  C

  $\frac{\sqrt{2} \mu_{0} \mathrm{i}}{8 \pi l}$
• D

  $\frac{\mu_{0} \dot{1}}{2 \sqrt{2} \pi l}$

Solution

Magnetic field due to finite length of a wire, $$ B=\frac{\mu_{0}}{4 \pi} \cdot \frac{i}{r}\left(\sin \phi_{1}+\sin \phi_{2}\right) $$
Here, $\phi_{1}=0^{\circ}, \phi=45^{\circ}$ $$ \begin{array}{l} B=\frac{\mu_{0}}{4 \pi} \cdot \frac{i}{r}\left(\sin 0^{\circ}+\sin 45^{\circ}\right)=\frac{\mu_{0}}{4 \pi} \cdot \frac{i}{\sqrt{2 l}} \\ \Rightarrow B=\frac{\sqrt{2} \mu_{0} i}{8 \pi l} \end{array} $$

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