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# Torque and potential energy of a dipole - Equilibrium position, Practice problems, FAQs

We know an electric dipole is like a dumble with two charges of the same quantity but opposite nature and separated by some distance. Have you ever thought how an electrical dipole will behave in an electric field? Will it move in the direction of the electric field, or opposite to it? Will it spin? Let’s find answers to these questions in this article!

Table of content:

• Force on an electrical dipole in a uniform electric field
• Torque on an electrical dipole in a uniform electric field
• Potential energy of an electrical dipole in a uniform electric field
• Equilibrium positions of an electrical dipole in a uniform electric field
• Practice problems
• FAQs

## Force on an electrical dipole in a uniform electric field

In a uniform electric field, the direction and magnitude of the electric field at every point is constant. This can be represented by parallel equidistant lines. A uniform electric field can be found between the plates of a parallel plate capacitor. Suppose an electric dipole of dipole moment  be placed in a uniform electric field, at a general angle θ. The force  will act on the positive charge and  on the negative charge. These two forces are equal in magnitude but in opposite directions. Thus, the net force,  on the dipole.

In case of a non-uniform electric field, the magnitude of force on the positive charge may not be the same as that on the negative charge. So, the net force on the dipole may or may not be zero.

## Torque on an electrical dipole in a uniform electric field

An electrical dipole in a uniform electric field doesn’t experience any net force but indeed experiences torque.

The torque due to the force on the positive charge, about the midpoint of the dipole is,

The torque due to the force on the negative charge about the midpoint of the dipole is, .

The net torque on the dipole is,

So,

The direction of the torque is perpendicular to the plane of the dipole.

It is minimum, equal to 0 when the dipole is parallel or antiparallel to the electrical field, i.e., θ=0° or 180°.

It is maximum when it is perpendicular to the electric field i.e., .

## Potential energy of an electrical dipole in a uniform electric field

An electric dipole when placed in an electric field experiences torque. To rotate the dipole through a small angle θ, the work done by the externally acting torque is, dW=-τ dθ= -PE sinθ dθ

The direction of the torque is counterclockwise but the direction of the angular displacement is clockwise. Thus, the work is negative as the rotation is against the torque.

So, the change in electrical potential energy of the dipole is -dW=PE sinθ dθ

This means if the dipole is rotated from an angle θ1 to θ2, the stored potential energy of the dipole

will be

In general,

## Equilibrium positions of an electrical dipole in a uniform electric field

When an electric dipole is placed in a uniform electric field, the net force on it is zero at any orientation of the dipole. But the torque acting on the dipole is not zero. Only at these two particular positions of θ=0° and 180° we can see, net force and torque on it is zero.

• When the dipole is in equilibrium position, at θ=0° , it is called the stable equilibrium position. Because the potential energy in this position is minimum. $P{E}_{0°}=-\stackrel{\to }{P}.\stackrel{\to }{E}=-PEcos0°=-PE$. When it is displaced slightly from this position of θ=0°, restoring torque brings it back to this equilibrium position.
• When the dipole is in the equilibrium position, at θ=180° , it is called the unstable equilibrium position. Because the potential energy in this position is maximum. PE180° $-\stackrel{\to }{P}.\stackrel{\to }{E}$ = - P.E=-PE cos 180°=+PE. When it is displaced slightly from this position of θ=180°, restoring torque will rotate it further from this equilibrium position.

## Practice problems:

Q. A dipole having dipole moment 8 μCm is placed in an electric field of 10 NC-1 i makes angle of +30° with the x axis. Calculate the torque experienced by the dipole.

A. Torque τ = PEsinθ  =8 × 10-6 × 10 × sin 30° = 40 × 10-6 Nm

Q. A dipole made of 2 μC charges separated by a distance 1 mm rests along the y axis as shown. What will be the dipole moment in vector form?

A. The magnitude of the dipole is 2 μC×1 mm=2×10-9 Cm.

The direction is always from the negative to the positive charge.

So P = 2 × 10-9 Cm (-j)

Q. A dipole of 10 μCm is under an electric field of 10 NC-1 i makes angle of +60° with the x axis. The dipole is made to rotate in CCW direction to make it aligned with the x axis . Calculate the change in potential energy of the dipole.

A. The potential energy of a dipole is a function of θ where θ is the angle between the dipole moment vector and the Electric field vector.

In general, \

So,

So, the change in potential energy will be,

= 1.5 .

Q. An electric field is present in the space as shown in the figure. At what angle with the +ve x axis the dipole will make the equilibrium. Which are stable and unstable equilibrium positions?

A. We know the equilibrium positions of a dipole are at the positions parallel and anti-parallel to the electric field. Parallel position is the stable equilibrium position. Anti-parallel position is the unstable equilibrium position.

Here, the electric field is at 45° with the +x axis.

So, the stable equilibrium position is when the dipole moment will make 45° with the positive x axis. The unstable equilibrium position is when the dipole moment will make 180°+45°=225° with the positive x axis.

## FAQs:

Q. Where is the potential energy of a dipole maximum?

A. Electrical dipole when placed in an electric field experiences torque. To rotate the dipole through a small angle θ, the work done by the torque is dW=-τ = -PE sinθ

The change in electrical potential energy of the dipole is ]. The stored potential energy of the dipole will be

In general,

So potential energy is maximum at  i.e.

Potential energy is minimum at θ = 0° i.e.

Numerically in terms of magnitude both are the same.

Q. A dipole is placed in an electrical field and the dipole is made to rotate in the field. At a position the orientation of the dipole can be described as the angle θ between the dipole moment and the electric field. By graph, describe the torque experienced by the dipole as varying with θ.

A. The net torque on the dipole can be expressed as

So, it will follow the sine curve.

Q. What are the stable and unstable equilibrium positions for a dipole in an electric field?
A. The equilibrium positions of a dipole are at the positions parallel and anti-parallel to the electric field. Parallel position is the stable equilibrium position. When it is displaced slightly from this position , a restoring torque by electric field brings it back to this equilibrium position.

Anti-parallel position is the unstable equilibrium position. When it is displaced slightly from this position , the torque will rotate it further from this equilibrium position.

Q. What is the direction of dipole moment, P ?
A. The direction of P is from the negative charge of a dipole to the positive charge of a dipole.

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