question_answer

Figure shows a straight wire of length l current i. The magnitude of magnetic field produced by the current at point P is

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

B) \[\frac{{{\mu }_{0}}i}{4\pi l}\]

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

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

Correct Answer: C

Solution :

The given situation can be redrawn as follow: As we know the general formula for finding 41 magnetic field due to a finite length wire \[B=\frac{{{\mu }_{0}}}{4\pi }\cdot \,\,\frac{i}{r}\,\,(sin\,{{\phi }_{1}}\,+\,sin\,{{\phi }_{2}})\] Here \[{{\phi }_{1}}=\,\,0{}^\circ ,\,\,\phi \,\,=\,\,54{}^\circ \] \[\therefore \,\,\,B=\frac{{{\mu }_{0}}}{4\pi }\cdot \frac{i}{r}\,(sin\,0{}^\circ \,+\,\sin \,45{}^\circ )\,\,=\,\frac{{{\mu }_{0}}}{4\pi }\,\cdot \,\,\frac{i}{\sqrt{2}l}\,\] \[\Rightarrow \,\,\,B=\frac{\sqrt{2}{{\mu }_{0}}i}{8\pi l}\]