question_answer

Two identical magnetic dipoles of magnetic moments 1.0 A-m2 each, placed at a separation of 2m with their axis perpendicular to each other. The resultant magnetic field at a point midway between the dipoles is        [Roorkee 1995]

A)            \[5\times {{10}^{-7}}T\]  

B)            \[\sqrt{5}\times {{10}^{-7}}T\]

C)            \[{{10}^{-7}}T\]                   

D)            None of these

Correct Answer: B

Solution :

                   With respect to 1st magnet, P lies in end side-on position                    \[\therefore {{B}_{1}}=\frac{{{\mu }_{0}}}{4\pi }\left( \frac{2M}{{{d}^{3}}} \right)\]      (RHS)                    With respect to 2nd magnet. P lies in broad side on position.                    \[\therefore \ {{B}_{2}}=\frac{{{\mu }_{0}}}{4\pi }\left( \frac{M}{{{d}^{3}}} \right)\]     (Upward)                    \[{{B}_{1}}={{10}^{-7}}\times \frac{2\times 1}{1}=2\times {{10}^{-7}}T,\ {{B}_{2}}=\frac{{{B}_{1}}}{2}={{10}^{-7}}T\]                    As B1 and B2 are mutually perpendicular, hence the resultant magnetic field            \[{{B}_{R}}=\sqrt{B_{1}^{2}+B_{2}^{2}}=\sqrt{{{(2\times {{10}^{-7}})}^{2}}+{{({{10}^{-7}})}^{2}}}\]           \[=\sqrt{5}\times {{10}^{-7}}T\]