Question

In the situation shown in the diagram, magnitude of q << | Q | and r>>a. The net force on the free charge -q and net torque on it about O at the instant shown are respectively
[ p = 2aQ is the dipole moment ]

• A. \(\frac{1}{4\pi\epsilon_0}\frac{pq}{r^2}\hat{k},\frac{1}{4\pi\epsilon_0}\frac{pq}{r^3}\hat{i}\)
• B. \(-\frac{1}{4\pi\epsilon_0}\frac{pq}{r^2}\hat{k},-\frac{1}{4\pi\epsilon_0}\frac{pq}{r^3}\hat{i}\)
• C. \(\frac{1}{4\pi\epsilon_0}\frac{pq}{r^3}\hat{i},+\frac{1}{4\pi\epsilon_0}\frac{pq}{r^2}\hat{k}\)
• D. \(\frac{1}{4\pi\epsilon_0}\frac{pq}{r^3}\hat{i},-\frac{1}{4\pi\epsilon_0}\frac{pq}{r^2}\hat{k}\)

Answer 1

The Correct Option is D

Given:

• Charge \( q \) is much smaller than \( |Q| \). 
• Distance \( r \gg a \).
• Dipole moment \( p = 2aQ \).

Step 1: Finding the Net Force on Charge \(-q\)

The force on a charge due to a dipole at a large distance \( r \) is given by:

\[ \mathbf{F} = \frac{1}{4\pi\epsilon_0} \frac{pq}{r^3} \hat{i} \]

Step 2: Finding the Net Torque about Point O

The torque on the charge \(-q\) due to the dipole is:

\[ \mathbf{\tau} = -\frac{1}{4\pi\epsilon_0} \frac{pq}{r^2} \hat{k} \]

Answer: The correct option is D.