question_answer

A charged ball B hangs from a silk thread S, which makes an angle \[\text{ }\!\!\theta\!\!\text{ }\] with a large charged conducting sheet P, as shown in the figure. The surface charge density \[\sigma \] of the sheet is proportional to

A)  \[\cot \,\text{ }\!\!\theta\!\!\text{ }\]                                 

B)  \[\cot \,\text{ }\!\!\theta\!\!\text{ }\]

C)  \[\tan \,\text{ }\!\!\theta\!\!\text{ }\]                                 

D)  \[\sin \,\text{ }\!\!\theta\!\!\text{ }\]

Correct Answer: C

Solution :

In equilibrium \[T\,\,\cos \,\theta =mg\]                            ?.(i)                 and        \[T\,sin\,\theta =F=Eq\]                                 \[T\,sin\,\theta =F=Eq\]               ?.(ii)                 \[\therefore \]  \[\frac{T\,\sin \,\theta }{T\,\cos \,\theta }=\frac{\sigma q}{2{{\varepsilon }_{0}}\times mg}\]                                 \[\tan \theta =\frac{\sigma q}{2{{\varepsilon }_{0}}mg}\]                 or            \[\sigma \propto \tan \theta \]