13.4 Dipole in an external electric field

  Electrostatics

The net force on an electric dipole in a uniform external electric field is zero. However, the dipole in the presence of an external electric field experiences a torque and has a tendency to align itself along the external electric field.

Torque on dipole $=$ force $ \times $ force arm

$$\tau = qE\left( {I\sin \theta } \right) = \left( {qI} \right)\left( {E\sin \theta } \right)$$$$\tau = pE\sin \theta $$

or,$$\vec \tau = \vec p \times \vec E$$

As $\overrightarrow E $ is a conservative field, work done by an external agent in changing the orientation of the dipole is stored as potential energy in the system of a dipole present in an external electric field.

$$W = \int {\tau \;d\theta \Rightarrow \quad \quad W = \int {pE\;\sin \;\theta \;d\theta = - pE\left[ {\cos \;\theta } \right]} } _{{\theta _1}}^{{\theta _2}}$$

We assume ${\theta _1} = 90^\circ $ as the datun for measuring potential energy can be chosen anywhere. $$ \Rightarrow U = - pE\;\cos \;\theta \quad or\quad U = - \vec p.\vec E$$

Question: Two tiny spheres, each of mass $M$ and having charges $+q$ and $-q$, are connected by a massless rod of length $L$. They are placed in a uniform electric field at an angle $\theta $ with $\overrightarrow E $ $\left( {\theta \approx 0} \right).$ Calculate the minimum time in which the system aligns itself parallel to $\overrightarrow E $

Solution: $$\tau = pE\;\sin \;\theta $$ $$ \Rightarrow \quad \tau = - \left( {pE} \right)\theta \left( {as\;\theta \to 0,\sin \;\theta \to \theta } \right)$$ $$ \Rightarrow \quad \alpha = - \left( {\frac{{pE}}{I}} \right)\theta = - {\omega ^2}\theta $$

As torque is proportional to $\theta $ and oppositely directed, therefore motion will be an S.H.M. Here, $p=qL$ and moment of inertia is, $$I = M{\left( {\frac{L}{2}} \right)^2} = \frac{{M{L^2}}}{2}$$ Therefore, time period is, $$T = 2\pi \sqrt {\frac{I}{{pE}}} $$

The minimum time required to align itself along electric field is $\frac{T}{{4.}}$