Question

In the figure there are four arcs carrying positive and negative charges. All of them have same charge density $\lambda$. Pick incorrect statement(s) :

• A. The net dipole moment for the given charge distribution is $\left(4\sqrt{5}\right)\lambda R^2$.
• B. The resultant electric filed at thee center is zero.
• C. If a uniform $\overrightarrow{E}$ is switched on perpendicular to the plane then charge distribution starts rotating about X-axis.
• D. Potential at the centre of the given charge distribution is non-zero.

Answer 1

Answer: (B), (C), (D)

The correct options are
B The resultant electric filed at thee center is zero.
C If a uniform $\overrightarrow{E}$ is switched on perpendicular to the plane then charge distribution starts rotating about X-axis.
D Potential at the centre of the given charge distribution is non-zero.

The electric field at the center is not zero as the vertical and horizontal components do not cancel out. (See the given diagram , no need to calculate).
The potential at the center is zero as the potential due to negative and positive charges cancel each other (Note: Potential is a scalar quantity whereas E is a vector quantity).
For dipole moment,
$P={\int }_0^{\pi /2}\lambda Rd\theta (Rcos\theta \hat{i}+Rsin\theta \hat{j})+{\int }_{\pi /2}^{\pi }\lambda rd\theta (rcos\theta \hat{i}+rsin\theta \hat{j})$
$-{\int }_{\pi }^{3\pi /2}\lambda Rd\theta (Rcos\theta \hat{i}+Rsin\theta \hat{j})-{\int }_{3\pi /2}^{2\pi }\lambda rd\theta (rcos\theta \hat{i}+rsin\theta \hat{j})$
$=\lambda R^2(2\hat{i}+2\hat{j})+\lambda r^2(-2\hat{i}+2\hat{j})$
$=4\lambda R^2\hat{i}+8\lambda R^2\hat{j}$
Since $\tau =\overrightarrow{p}\times \overrightarrow{E}$, torque is not in i direction.