A charged ball B hangs from a silk thread S, which makes an angle \[\theta \] 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 \[\theta \]
B) cot \[\theta \]
C) tan \[\theta \]
D) sin \[\theta \]
Correct Answer:
C
Solution :
In equilibrium \[T\cos \theta =mg\] ... (i) and \[T=\sin \theta =F=Eq\] \[T\sin \theta =\frac{\sigma }{2{{\varepsilon }_{0}}}\cdot q\] ... (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 \[\sigma \propto \tan \theta \]