Question

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. $cos\,\theta$
• B. $cot\,\theta$
• C. $sin\,\theta$
• D. $tan\,\theta$

Answer 1

The Correct Option is D

Electric field due to a charged conducting sheet of surface charge density $\sigma$ is given by $E=\frac{\sigma}{\varepsilon_{0}\varepsilon_{r}}$ where, $\varepsilon_{0}=$ permittivity in vacuum and $\varepsilon_{r}$ relative permittivity of medium. Here, electrostatic force on $B$ $QE=\frac{Q\,\sigma}{\varepsilon_{0}\varepsilon_{r}}$ $FBD$ of $B$ is shown in figure. In equilibrium, $T \,cos\, \theta = mg$ and $T\,sin\,\theta=\frac{Q\,\sigma}{\varepsilon^{\cdot}_{0}\varepsilon_{r}}$ Thus, $tan\,\theta\,\propto\,\sigma$ or $\sigma\,\propto\,tan\,\theta$