Question

An electric dipole with charges 2µC and separation 20 cm is placed closed to an infinitely charge non-conducting sheet with surface charge density 100 cm². Find the torque acting on the dipole if the dipole makes an angle 30º with the formal to the sheet?

• A. $\frac{12}{\epsilon_0}$ x 10-5 nm
• B. $\frac{2}{\epsilon_0}$ x 10-5 nm
• C. $\frac{4}{\epsilon_0}$ x 10-5 Nm
• D. $\frac{1}{\epsilon_0}$ x 10-5 Nm

Answer 1

Answer: (D)

$\frac{1}{\epsilon_0}$ x 10-5 Nm

Answer 2

Solution

1. The dipole moment is

   $p = q\,d = (2\times10^{-6}\,C)(0.2\,m)= 4\times10^{-7}\,C\cdot m.$

2. The electric field due to an infinite, uniformly charged sheet is

   $E=\frac{\sigma}{2\epsilon_0}\,.$

   Assuming the given surface charge density is $\sigma=100\,C/m^2$ (so that units and answer options become consistent), we have

   $E=\frac{100}{2\epsilon_0} = \frac{50}{\epsilon_0}\,.$

3. The torque on a dipole in a uniform electric field is

   $\tau = pE\sin\theta\,.$

   For $\theta=30^\circ$ (with $\sin30^\circ=\frac12$),

   $\tau= 4\times10^{-7}\times \frac{50}{\epsilon_0}\times\frac{1}{2}=\frac{4\times 50}{2}\times\frac{10^{-7}}{\epsilon_0}=\frac{100}{\epsilon_0}\times10^{-7}=\frac{1}{\epsilon_0}\times10^{-5}\,Nm.$

Explanation (minimal):

• Compute dipole moment: $p=2\times10^{-6}\times0.2=4\times10^{-7}\,Cm$.
• For an infinite sheet, $E=\sigma/(2\epsilon_0)=50/\epsilon_0$ (taking $\sigma=100\,C/m^2$).
• Use torque formula: $\tau=pE\sin30^\circ=4\times10^{-7}\times(50/\epsilon_0)\times(1/2)= (1/\epsilon_0)\times10^{-5}\,Nm$.