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 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 (0) formed between the line $A A^{\prime}$ and the string is

• A

  $\frac{q \sigma}{2 \varepsilon_{0} m g}$
• B

  $\frac{q \sigma}{4 \pi \varepsilon_{g} m g}$
• C

  $\frac{q \sigma}{2 \pi e_{0} m g}$
• Correct answer
  D

  $\frac{q \sigma}{\varepsilon_{0} m g}$

Solution

The diagram is as follows The electric field due to charged infinite conducting sheet is $E=\frac{\sigma}{\varepsilon_{0}}$ Now, force (electric force) on the charged ball is $F=q E=\frac{q \sigma}{\varepsilon_{0}}$ The resultant of electric force and $m g$ balance the tension produced in the string. So, tan $\theta=\frac{F_{e}}{m g}=\frac{q \sigma}{\frac{\varepsilon_{0}}{m g}}=\frac{q \sigma}{\varepsilon_{0} m g}$

Hence, the correct answer is option: 'D'.