question_answer

Figure shows a straight wire of length \[l\] carrying 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 follows. As we know the general formula for finding 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, \[{{\phi }_{1}}={{0}^{o}},\,\,\phi ={{45}^{o}}\] \[\therefore \]   \[B=\frac{{{\mu }_{0}}}{4\pi }.\frac{i}{r}\,(\sin \,{{0}^{o}}+\sin {{45}^{o}})\] \[=\frac{{{\mu }_{0}}}{4\pi }.\frac{i}{\sqrt{2}l}\] \[\Rightarrow \] \[B=\frac{\sqrt{2}{{\mu }_{0}}i}{8\pi l}\]