Electrostatic Dipole

ELECTRIC DIPOLE

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

Figure represents an electric dipole consisting of two charge – q and + q and separated by distance 2a. This distance is called length of the dipole and is a vector , whose direction is from the charge –q to charge +q.

Electric dipole moment

It is defined as the product of either charge and the length of the electric dipole.

It is denoted by vector $\overrightarrow{p}$ , which has the same direction as that of $2a$ . Thus,

$\overrightarrow{p}=q\left(2\overrightarrow{a}\right)$

Unit of Dipole moment is coulomb metre (C m).

Electric field on axial line of an electric dipole

Consider an electric dipole consisting of charges – q and + q, separated by a distance 2a and placed in free space. Let P be a point on the line joining the two charges (axial line) at a distance r from the centre O of the dipole.

The electric field at point P due to the dipole will be the resultant of the electric fields (due to charge –q at the point A) and (due to charge +q at the point B)i.e.

Now, q (2a) = p, the magnitude of the electric dipole moment of the dipole.

It may be noted that direction of electric field at a point on axial line of the dipole is from charge –q to +q i.e. same as that of electric dipole moment of the dipole. Therefore, in vector notation,

When dipole is of very small length

If the dipole is of small length, such that a² can be neglected as compared to r². Therefore, for an electric dipole of very small length

Electric field on equatorial line of an electric dipole

Consider an electric dipole consisting of charges –q and +q separated by a distance 2a and placed in free space. Let P be a point on equatorial line of the dipole (right bisector of the length of dipole) at a distance r from the centre of the dipole.

Let $\underset{E_A}{\to }$ and $\underset{E_B}{\to }$be electric field at point P due to charge –q at point A and charge +q at point B. Then, resultant electric field at point P is given by

$\overrightarrow{E}=\overrightarrow{E_A}+\overrightarrow{E_B}$

It may be noted that direction of electric field at a point on the equatorial line of the dipole is from charge +q to –q i.e. opposite to the direction of electric dipole moment of the dipole. Therefore, in vector notation,

When dipole is of very small length

If the dipole is of small length, such that a² can be neglected as compared to r². Therefore, for an electric dipole of very small length,

E = $\frac{1}{4{\mathrm{\pi \xi }}_0}$. $\frac{2p}{r^3}$(along PX)

Torque on a dipole in a uniform electric field

Consider an electric dipole consisting of charges – q and +q and of length 2a placed in a uniform electric field making an angle θ with the direction of the field as shown in figure.

Force on charge – q at A = - q $\overrightarrow{E}$ (opposite to $\overrightarrow{E}$ )

And force on charge + q at B = q $\overrightarrow{E}$ (along $\overrightarrow{E}$ )

Thus, electric dipole is under the action of two equal and unlike parallel forces, which gives rise to a torque on the dipole. The magnitude of the torque is given by

$\zeta$ = either force perpendicular distance between the two forces

= q E (AN) = q E (2 a sin θ) = q (2a) E sin θ

Or $\zeta$ = p E sin θ

Here, p = q (2a) is electric dipole moment of the electric dipole

The torque on the dipole tends to align it along the direction of the electric field.

Since electric dipole moment vector is a vector, the equation may be expressed as

$\zeta =\underset{p}{\to }\times \underset{E}{\to }$

Note:

1. It may be pointed out that when dipole is placed in uniform electric field, it experiences only a torque. Net force on the dipole is zero.

2. Torque on the dipole becomes zero, when it aligns itself parallel to the electric field.

3. Torque on the dipole is maximum, when dipole is placed at right angles to the direction of the electric field.

${\zeta }_{max}=pE\sin {90}^0=pE$