question_answer

Two identical short bar magnets, each having magnetic moment M, are placed a distance of 2d apart with axes perpendicular to each other in a horizontal plane. The magnetic induction at a point midway between them is [IIT-JEE (Screening) 2000]

A)            \[\frac{{{\mu }_{0}}}{4\pi }(\sqrt{2})\frac{M}{{{d}^{3}}}\] 

B)            \[\frac{{{\mu }_{0}}}{4\pi }(\sqrt{3})\frac{M}{{{d}^{3}}}\] 

C)            \[\left( \frac{2{{\mu }_{0}}}{\pi } \right)\,\frac{M}{{{d}^{3}}}\]       

D)            \[\frac{{{\mu }_{0}}}{4\pi }(\sqrt{5})\frac{M}{{{d}^{3}}}\]

Correct Answer: D

Solution :

           At point P net magnetic field \[{{B}_{net}}=\sqrt{B_{1}^{2}+B_{2}^{2}}\]                    where \[{{B}_{1}}=\frac{{{\mu }_{0}}}{4\pi }.\frac{2M}{{{d}^{3}}}\]and \[{{B}_{2}}=\frac{{{\mu }_{0}}}{4\pi }.\frac{M}{{{d}^{3}}}\]                    \[\Rightarrow {{B}_{net}}=\frac{{{\mu }_{0}}}{4\pi }.\frac{\sqrt{5}M}{{{d}^{3}}}\]