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) \[{{\left( 2\,\,tan\,\,\theta  \right)}^{1/3}}\]                     

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

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

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

Correct Answer: C

Solution :

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