question_answer

Let a straight wire of length I carries a current\[i\]. The magnitude of magnetic field produced by the current at point P (as shown in figure) is

A)  \[\frac{{{\mu }_{0}}i}{2\sqrt{2}\pi l}\]         

B)  \[\frac{\sqrt{2}{{\mu }_{0}}i}{8\pi l}\]

C)  \[\frac{{{\mu }_{0}}i}{4\pi l}\]

D)  \[\frac{\sqrt{2}{{\mu }_{0}}i}{\pi l}\]

Correct Answer: B

Solution :

The magnetic field due to a finite length wire                 \[B=\frac{{{\mu }_{0}}}{4\pi }.\,\frac{i}{r}\,(\sin {{\phi }_{1}}+\sin {{\phi }_{2}})\] Here,     \[{{\phi }_{1}}={{0}^{o}}\] and \[{{\phi }_{2}}={{45}^{o}}\] \[\therefore \]  \[B=\frac{{{\mu }_{0}}i}{4\pi }\frac{i}{r}\,(\sin \,{{0}^{o}}+\sin \,{{45}^{o}})\]                 \[=\frac{{{\mu }_{0}}}{4\pi }\frac{i}{l}=\frac{1}{\sqrt{2}}\] \[=\frac{\sqrt{2}{{\mu }_{0}}i}{8\pi l}\]