Q.
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:
AIEEEAIEEE 2005Electric Charges and Fields
Solution:
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 $\sigma\,\propto\,tan\,\theta$
