Question

Calculate the force and torque acting on an electric dipole placed in an external uniform electric field, and define dipole moment and explain when the torque is maximum and minimum.

Answer 1

Electric dipole and its units: 

Electric dipole. A system of two equal and opposite charges separated by certain distance is called electric dipole.

Fig. shows an electric dipole consisting of two equal and opposite point charges (± q) separated by a small distance 2a.

Dipole moment (\(\vec P\)). It is defined as the product of magnitude of either charge and the distance between two charges.  

i.e. \(\vec P= q \vec{2a}\)

The direction of \(\vec p\) is from -ve charge to +ve charge.
Units: In S.I.,

The unit of \(\vec p\) is Coulomb-metre (Cm).

Dipole field. The electric field produced by an electric dipole is called electric dipole field and is defined as the space around an electric dipole within which the effect of the dipole can be felt.

Electric lines of force. Below Fig. shows the section of the field in the plane of paper, containing dipole itself.

The lines of force are directed away from +q charge and towards -q charge. They are straight and open along the line containing the charges. This line is called axial line of dipole. On the broad side of the dipole, lines are curved. Near the dipole, they start from +q charge and end at -q charge, then closing their path at the ends of the dipole.

Expression for torque

Consider a uniform electric field having constant magnitude and direction the same at all planes. Consider an electric dipole with its electric dipole moment vector, \(\vec p\)  making an angle θ with the field. (Fig.)

Since \(\vec E\) at any point is the force experienced per unit charge.

Hence, in a uniform electric field an electric field dipole experiences no net force; and therefore dipole does not undergo any translatory motion.
Torque acting on the dipole
These forces are equal in magnitude and opposite in direction, therefore, they form a couple.

From eq. (4), we have

i.e. \(|\vec \tau|\) = pE sin θ

(∴ qa = p = Electric Dipole Moment)

The forces acting on dipole, try to rotate the dipole clockwise and according to right hand rule of rotation, the torque acts downwards ⊥ to the plane containing \(\vec p\) and \(\vec E\)

i.e. \(\vec \tau \ \bot \ \vec p\)

or \(\vec \tau \ \bot \ \vec E\).....................(5)

The equations (4) and (5) satisfy the condition for cross-product of vector

∴ Either \(\vec \tau= \vec p\ \times\ \vec E\) ........................(6)

or \(\vec \tau= \vec p\ \times\ \vec E\) ........................(7)

According to right-hand rule of vector product of vectors, only the equation (6) is satisfactory.

Hence, the accepted relation is

\(\vec \tau= \vec p\ \times\ \vec E\)

Thus, it is seen that in a uniform electric field, dipole experiences only a torque but no net force.

Maximum torque acts when θ = 90°

So τ_max = P E sin 90° = pE

Minimum torque acts when θ = 0°

τ_min = pE Sin 0° = 0

Definition of dipole moment

Since τ = pE sin θ

So if E = 1, θ = 90°, then

τ = p

or p = τ

i.e. dipole moment of an electric dipole is the torque acting on a dipole, placed at right angle to the uniform electric field of unit strength.