question_answer

Two magnets A and B are identical and these are arranged as shown in the figure. Their length is negligible in comparison to the separation between them. A magnetic needle is placed between the magnets at point P which gets deflected through an angle \[\theta \] under the influence of magnets. The ratio of distance \[{{d}_{1}}\] and \[{{d}_{2}}\] will be

A)            \[{{(2\tan \theta )}^{1/3}}\]  

B)            \[{{(2\tan \theta )}^{-1/3}}\]

C)            \[{{(2\cot \theta )}^{1/3}}\]

D)            \[{{(2\cot \theta )}^{-1/3}}\]

Correct Answer: C

Solution :

                   In equilibrium \[{{B}_{1}}={{B}_{2}}\tan \theta \] \[\Rightarrow \frac{{{\mu }_{0}}}{4\pi }.\frac{2M}{d_{1}^{3}}=\frac{{{\mu }_{0}}}{4\pi }.\frac{M}{d_{2}^{3}}\tan \theta \]            \[\Rightarrow \frac{{{d}_{1}}}{{{d}_{2}}}={{\left( 2\cot \theta  \right)}^{1/3}}\]