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 ACBD is                                                                          [MP PMT 1994]

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