Question

Maximum torque acting on an electric dipole of moment $3\times {{10}^{-29}}Cm$ in a uniform electric field $E$ is $6\times {{10}^{-25}}Nm$. Find $E$.

Answer 1

When an electric dipole is kept in an electric field, force acts on it due to the electric field. The force acts on it in such a way that it wants to bring the dipole in an equilibrium position. The most stable position is when the dipole makes ${{0}^{o}}$ and the most unstable position is when it makes ${{90}^{o}}$ .

Formula used: $\tau =pE\sin \theta$

Complete step by step solution:
An electric dipole is made up of two electric charges of equal magnitudes but opposite signs. Electric dipole moment is defined as the product of charge on the dipole and the distance between them. Its direction is from the positive charge to the negative charge. Its SI unit is $C\,m$
When an electric dipole moment is kept in an electric field, force acts on both charges in a dipole. The force acting on the charges acts as a couple and the dipole experiences a torque.
The torque is maximum when it makes ${{90}^{o}}$ with the electric field, which is the most unstable position.
Torque acting on a dipole is given by-
$\tau =pE\sin \theta$ ---------- (1)
Here,
$\tau$ is the torque acting on dipole due to the electric field
$p$ is the dipole moment
$E$ is the magnitude of electric field
$\theta$ is the angle between the dipole and electric field.
Substituting given values in eq (1), we get,
$6\times {{10}^{-25}}=3\times {{10}^{-29}}\times E$ [ $\theta ={{90}^{o}}$ as torque is maximum]
$2\times {{10}^{4}}C\,{{m}^{2}}=E$

Therefore, the electric field for maximum torque acting on the dipole is $2\times {{10}^{4}}C\,{{m}^{2}}$ .

Note: Electric field is defined as the force acting per unit charge. It is given as $E=\dfrac{kq}{{{r}^{2}}}$ ( $k$ is constant, $q$ is the magnitude of charge which exerts its force on the unit charge, $r$ is the distance between charge $q$ and the unit charge). Electric lines of forces are lines originating from the charge which indicate the direction of the field.