question_answer

Two magnets A and B are identical and these are arranged as shown. Their lengths are negligible in comparison to separation between them. A magnetic needle is placed between the magnets at point P and it gets deflected through an angle \[\theta .\] The ratio of distances  and will be

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

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

C) \[(2\cot \theta )\]            

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

Correct Answer: A

Solution :

Needle will deflect to magnetic field direction at P

At P, B is produced by magnet A and B

B due to magnet \[B={{B}_{1}}\]

B due to magnet \[A={{B}_{2}}\]

Formula of B: At axis of magnet \[A=\frac{{{\mu }_{0}}M}{{{d}^{3}}}\]

At equatorial axis of magnet \[B=\frac{2{{\mu }_{0}}M}{{{d}^{3}}}\]

\[\therefore {{B}_{2}}=\frac{{{\mu }_{0}}M}{d_{2}^{3}}\]

\[{{B}_{1}}=\frac{2{{\mu }_{0}}M}{d_{1}^{3}}\]

\[\tan \,\,(90=\theta )=\frac{{{B}_{2}}}{{{B}_{1}}}\]

\[\frac{{{B}_{2}}}{{{B}_{1}}}=\cot \theta \]

\[\frac{1}{2}{{\left( \frac{{{d}_{1}}}{{{d}_{2}}} \right)}^{3}}=\cot \theta \]

\[\frac{d}{{{d}_{2}}}={{(2\cot \theta )}^{1/3}}\]