question_answer

Let a straight wire of length \[l\] 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 \[{{\phi }_{2}}={{45}^{o}}\] \[\therefore \]  \[B=\frac{{{\mu }_{0}}}{4\pi }\frac{i}{r}(sin{{0}^{o}}+sin{{45}^{o}})\] \[=\frac{{{\mu }_{0}}}{4\pi }\frac{i}{l}\frac{1}{\sqrt{2}}\] \[=\frac{\sqrt{2}{{\mu }_{0}}i}{8\pi l}\]