Electric Dipole and Electric Field due to Dipole

An electric dipole is a pair of equal and opposite point charges that are separated by a short distance.

Electric Dipole Moment

The electric dipole moment is the product of the magnitude of one electric charge and the distance between the charges.SI unit of dipole moment is Coulomb-meter.

Electric dipole formula,

\[p=(2a)q\]

p= Dipole moment, 2a = distance between the charges, q=Charge

The electric dipole moment is a vector. Its direction is from the negative charge to the positive charge.

Dipole field or The field of an electric dipole

Dipole field is the electric field produced by an electric dipole. The total charge of the electric dipole is zero but dipole field is not zero.

The dipole field has a cylindrical symmetry. The axis of the cylinder passed through the dipole axis.

The electric field at an axial point of an electric dipole

The line joining the two charges of the dipole is called axial line or axis of the dipole.

For a short dipole, r >> a,

\[E = \frac{2p}{4\pi\epsilon_{o}r^3}\]

E= Electric field, r=distance between the center of dipole and the point of observation., p=dipole moment.

The direction of E and is same as p.

The electric field at an equatorial point of an electric dipole

The line passing through the midpoint of axial line and perpendicular to the axial line is called equatorial line. It is also called normal bisector of the dipole.

For a short dipole, r >> a,

\[E = \frac{-p}{4\pi\epsilon_{o}r^3}\]

E=Electric field due to the dipole, r=distance from the midpoint of the dipole to the point of observation, p=dipole moment.

The electric field at the position (r,θ)

\[E=\frac{1}{4\pi\epsilon_{o}}\frac{p}{r^3}\sqrt{1+3\cos^2\theta}\]

$\theta$ is the angle formed by a line joining the point to the center of the dipole with dipole axis.

Direction of resultant intensity,

\[tan \  \alpha = \frac{1}{2} \ tan \ \theta\]

Case:1 When the point of observation lies on the axial line of dipole, $\theta=0^{\circ}, cos\ \theta = cos \ 0^{\circ}=1$.

\[E=\frac{p}{4\pi \epsilon_{o}}\sqrt{3 \cos^{2} \ 0^{\circ} + 1}=\frac{2p}{4 \pi \epsilon_{o}r^{3}}\]

\[tan \ \alpha = \frac{1}{2} \ tan \ 0^{\circ}=0 ,\alpha=0^{\circ}\],

i.e.The resultant intensity is along the axial line.

Case:2 When the point of observation lies on equatorial of dipole, $\theta=90^{\circ}, cos\ \theta = cos 90^{\circ}=0$.

\[E=\frac{p}{4\pi \epsilon_{o}}\sqrt{3 \cos^{2} \ 90^{\circ} + 1}=\frac{p}{4 \pi \epsilon_{o}r^{3}}\]

\[tan \alpha = \frac{1}{2} \ tan \ 90^{\circ}=\infty ,\alpha=90^{\circ}\],

i.e. The direction of resultant intensity is perpendicular to the equatorial line (and hence parallel to axial line of dipole).

Force on Electric Dipole

The net force on a dipole in the uniform electric field is zero it experiences only torque. While in a non-uniform electric field, it may or may not be zero.when the electric dipole is kept in the non-uniform electric field it experiences a torque but no force.

Read – Force between the charges (Coulomb’s Law)

Torque on an electric dipole

\[\text{Magnitude of torque }\tau=p\ E\ sin\ \theta\]

The direction of the torque is perpendicular to the plane of paper inwards.

Units of $\tau$ are N-m and its dimensional formula is [M1L2T-2].

Torque at different angles,

when$\theta=0^{\circ}$or $180^{\circ}$, the torque is zero, means when the dipole is parallel or anti-parallel to E.

AT,$\theta=90^{\circ}$- torque is maximum.

The work done in rotating an electric dipole in a uniform electric field

When the electric dipole is placed in an electric field E, a torque $\tau=p\times E$ acts on it.This torque tries to rotate the dipole through an angle $\theta$.

If the dipole is rotated from an angle $\theta_{1}$ to $\theta_{2}$, then work done by the external force is given by,

\[W=-pE(cos\ \theta_{1} – cos \ \theta_{2})\]

The work done is stored in the dipole in the form of energy, which is called the potential energy of the dipole. The potential energy of dipole is a scalar quantity. It is measured in joule.

If an electric dipole of moment $p$ is placed normal to the line of force of electric intensity E, then the work is done in deflecting it through an angle of $180^{\circ}$ is $- 2pE$.

A molecule with a dipole moment $p$ is placed in an electric field of strength $E$. Initially, the dipole is aligned parallel to the field. If the dipole is to be rotated to be anti-parallel to the field, Then the work required to be done by an external agency is $- 2pE$.