Question

The line $A{A}^{\prime }$ is on charged infinite. $A$ conducting plane which is perpendicular to the plane of the paper. The plane has a surface density of charge $\sigma$ and $B$ is ball of $A^{\prime }$ mass $m$ with a like charge of magnitude q. $B$ is connected by string from a point on the line $A{A}^{\prime }$. The tangent of angle $(\theta )$ formed between the line $A{A}^{\prime }$ and the string is

• A. $\frac{q\sigma }{2{\epsilon }_{0}mg}$
• B. $\frac{q\sigma }{4\pi {\epsilon }_{0}mg}$
• C. $\frac{q\sigma }{2\pi {\epsilon }_{0}mg}$
• D. $\frac{q\sigma }{{\epsilon }_{0}mg}$

Answer 1

Answer: (D)

The diagram is as follows

The electric field due to charged infinite conducting sheet is
$E=\frac{\sigma }{{\epsilon }_0}$.
Now, force (electric force) on the charged ball is
$F=qE=\frac{q_0}{{\epsilon }_0}$
The resultant of electric force and $mg$ balance the tension produced in the string. So, $\tan \theta =\frac{F_{\theta }}{mg}$
$=\frac{q\sigma }{\frac{{\epsilon }_0}{mg}}=\frac{q\sigma }{{\epsilon }_{0}mg}$