Question

Given a square frame of diagonal length 2r made of insulating wires. There is a short dipole, having dipole moment P, fixed in the plane of the figure lying at the center of the square, making an angle $\theta$ as shown in figure. Four identical particles having charges of magnitude q each and alternatively positive and negative sign are placed at the four corners of the square. Select the correct alternative(s).

• A. Electrostatic force on the system of four charges due to dipole is $\frac{6kPq}{r^3}$
• B. Electrostatics force on the system of four charges due to dipole is $\frac{6kPq}{r^3} \cos \theta$
• C. Net torque on the system of four charges about the centre of the square due to dipole is zero
• D. Net torque on the system of four charges about the centre of the square due to dipole is $\frac{3kPq}{r^2}$

Answer 1

Answer: (A), (C)

A, C

Answer 2

The electric field at a position $\vec{r}$ due to a short dipole $\vec{P}$ at the origin is given by:

$\vec{E}(\vec{r}) = \frac{k}{|\vec{r}|^3} [3(\vec{P} \cdot \hat{r}) \hat{r} - \vec{P}]$

The force on the i-th charge is $\vec{F}_i = q_i \vec{E}(\vec{r}_i)$. The total force on the system of four charges is:

$\vec{F}_{total} = \sum_{i=1}^4 \vec{F}_i = \sum_{i=1}^4 q_i \vec{E}(\vec{r}_i)$.

After calculations, the electrostatic force on the system of four charges due to the dipole is found to be $\frac{6kPq}{r^3}$.

The net torque on the system of four charges about the center of the square due to the dipole is:

$\vec{\tau}_{total} = \sum_{i=1}^4 \vec{\tau}_i = \sum_{i=1}^4 \vec{r}_i \times \vec{F}_i = \sum_{i=1}^4 q_i (\vec{r}_i \times \vec{E}(\vec{r}_i))$.

After calculations, the net torque is found to be zero.