Question

An electric dipole is placed at an angle of ${30^0}$ with an electric field intensity $2 \times {10^5}N/C$ It experiences a torque equal to $4Nm$. The charge on the dipole, if the dipole is length is $2cm$, is
(A) $7\mu C$
(B) $8mC$
(C) $2mC$
(D) $5\mu C$

Answer 1

Hint: We know that if we place a dipole in a uniform electric field, the dipole experiences a torque. The angle, dipole length and torque and electric field intensity are given. We can use these in the equation of torque to find the charge.

Formula used:
\[\overrightarrow \tau = \overrightarrow p \times \overrightarrow E \]

Complete step by step answer
If we place a dipole of charge +q and -q of length say l and dipole moment vector$\overrightarrow p $ in a uniform electric field $\overrightarrow E $, then we observe that the force on +q will be $q\overrightarrow E$ and this will act in the direction of electric field and the force on -q will be $q\overrightarrow E $and this will act in the opposite direction of electric field. Since these forces are equal and opposite, thus there will be no net force on the dipole in an electric field if the dipole is aligned in the direction of the field.
If we displace the dipole by an angle $\theta $, then the force will act on different points and hence it will form a restoring couple which will force the dipole to be parallel to the electric field. Moment of couple observed in the dipole is the torque$\tau $ and is given by the equation
\[\overrightarrow \tau = \overrightarrow p \times \overrightarrow E = pE\sin \theta \]
substituting the given values of torque, angle and electric field we get,
\[4Nm = p \times 2 \times {10^5} \times \dfrac{1}{2} \Rightarrow p = 4 \times {10^{ - 5}}\]
Now, we know that dipole moment is given by $p = q \times l \Rightarrow q = 2 \times {10^{ - 3}} = 2mC$

Hence the correct option is C.

Note:
When the dipole is placed in the direction perpendicular to that of the electric field, there will be maximum torque acting on it. If you place a dipole in a non-uniform electric field, it will experience a torque and a force as well.