question_answer

Two straight long conductors AOB and COD are perpendicular to each other and carry currents \[{{i}_{1}}\] and \[{{i}_{2}}\]. The magnitude of the magnetic induction at a point P at a distance a from the point O in a direction perpendicular to the plane ABCD is

A) \[\frac{{{\mu }_{0}}}{2\pi a}({{i}_{1}}+{{i}_{2}})\]               

B) \[\frac{{{\mu }_{0}}}{2\pi a}({{i}_{1}}-{{i}_{2}})\]

C) \[\frac{{{\mu }_{0}}}{2\pi a}{{(i_{1}^{2}+i_{2}^{2})}^{1/2}}\]      

D) \[\frac{{{\mu }_{0}}}{2\pi a}\,\frac{{{i}_{1}}{{i}_{2}}}{({{i}_{1}}+{{i}_{2}})}\]

Correct Answer: C

Solution :

At \[P:{{B}_{net}}=\sqrt{B_{1}^{2}+B_{2}^{2}}\]\[=\sqrt{{{\left( \frac{{{\mu }_{0}}}{4\pi }\frac{2{{i}_{1}}}{a} \right)}^{2}}+{{\left( \frac{{{\mu }_{0}}}{4\pi }\frac{2{{i}_{2}}}{a} \right)}^{2}}}\]\[=\frac{{{\mu }_{0}}}{2\pi a}\,{{(i_{1}^{2}+i_{2}^{2})}^{1/2}}\]