question_answer

A magnetic dipole is under the influence of two magnetic fields. The angle between the field directions is \[60{}^\circ \] and one of the fields has a magnitude of \[1.2\times {{10}^{-2}}T\]. If the dipole comes to stable equilibrium at an angle of \[30{}^\circ \] with this field, then the magnitude of the field is:

A) \[1.2\times {{10}^{-4}}T\]                   

B)             \[2.4\times {{10}^{-4}}T\]

C)             \[1.2\times {{10}^{-2}}T\]                   

D)             \[2.4\times {{10}^{-2}}T\]

Correct Answer: C

Solution :

(c) \[1.2\times {{10}^{-2}}T\] Here, \[\theta =60{}^\circ ,\text{ }{{B}_{1}}=1.2\times {{10}^{-2}}T\] \[{{\theta }_{1}}=30{}^\circ \] and \[{{\theta }_{2}}=60{}^\circ -30{}^\circ =30{}^\circ \]                      In stable equilibrium, torques due to two fields must be balanced i.e. \[{{\tau }_{1}}={{\tau }_{2}}\] \[\Rightarrow M{{B}_{1}}\sin {{\theta }_{1}}=M{{B}_{2}}\sin {{\theta }_{2}}\] Or         \[{{B}_{2}}={{B}_{1}}\frac{\sin {{\theta }_{1}}}{\sin {{\theta }_{2}}}\]             \[={{B}_{2}}\frac{\sin 30{}^\circ }{\sin 30{}^\circ }={{B}_{1}}\]             \[=1.2\times {{10}^{-2}}T\]