If a dipole is in a uniform electric field, it will experience some force and a rotating effect. This rotating effect is known as ‘torque’. Torque is a vector quantity, and its direction generally depends only on the force applied. The torque is calculated based on the rotational effect experienced by the electric dipole as a result of its interaction with the electric field. It arises from the alignment of the dipole moment, which is a vector pointing from the negative to the positive charge, with the direction of the electric field.

In this article we will learn about the topic dipole in a uniform external field but first revise the basic terms like “Torque”, “Dipole”, “electric field”, and dipole in a uniform external field study material.

What is Dipole?

An electric dipole comprises of two equal and opposite point charges separated by a small distance. This configuration generates an electric field, which is characterized by its dipole moment. The dipole moment is a vector quantity pointing from the negative to the positive charge, with magnitude equal to the product of the charge magnitude and the distance between them.

What is Electric Field?

Electric field is the region around an electric charge where it influences other charges. It is a vector field, meaning it has both magnitude and direction at every point in space. The strength of the electric field at a particular point is defined as the force experienced by a unit positive charge placed at that point. The Electric Field is given as

E = F/q

where,

• E is electric field
• F is force experienced by a test charge
• q is magnitude of test charge

The direction of the electric field at a point is the same as the direction of the force experienced by a positive test charge placed at that point. For positive charges, the electric field lines point away from the charge, while for negative charges, the electric field lines point toward the charge.

Dipole in a Uniform External Field

Let us take a pair of electric charges of opposite sign but equal magnitude, separated by a very smaller distance. This is called dipole.

The arrangement of charges in the dipole causes dipole moment which is given as

p = q × d

where,

• q is charge of dipole
• d is distance between two charges.

Since the impact of an external electric field on charges is already known to us; a dipole will also experience some form of force when introduced to an external field. It is known that, a dipole placed in an external electric field possess a rotating effect. This term is known as ‘torque’. The net torque can be calculated on the opposite charges present in a dipole for estimating the overall rotation.

Torque on Electric Dipole in External Field

Assume a dipole that has the charges: +q and –q that form a dipole because a distance of d separates them. In this case, the dipole should be placed in the electric field that is uniform and has strength, represented by E.

The dipole’s axis makes an angle θ with the electric field. The torque experienced by dipole in uniform external

???? = p⨯E

or

???? = pE sin θ

where

• θ denotes the angle between p and E.
• p is dipole moment
• E is electric field

Now we know that dipole moment is product of charges and the distance between them. Let’s assume the magnitude of charge is q and they are separated by distance “2a”

In this case, p = q×2a

Therefore,

Torque (????) = p E sin θ = q E × 2a sin θ

Observations in Net Force and Torque

• If dipole moment and external field are parallel or antiparallel i.e. angle between p and E is zero, no torque will be experienced as sin 180 = 0
• If the external field is non-uniform then net force can never be zero hence dipole will always experience torque
• The force and torque on dipole depends on the orientation of the dipole

Potential Energy on Dipole in External Field

Consider a dipole with charges +q and -q placed in a uniform electric field as shown in the figure above. The charges are having a distance d and E be the magnitude of an electric field. The force experienced by the charges is given as –qE and +qE, as can be seen in the figure.

As we know that, when a dipole is placed in a uniform electric field, both the charges completely not experience any force, but it experiences a torque equal to t which is given as

Torque = p⨯E

This torque turn the dipole that is placed parallel /anti-parallel to the field. If we apply an external and opposite torque, it neutralizes the effect of this torque given by external torque and it rotates the dipole from the angle θ₀ to an angle θ₁ at an very small angular speed without any angular acceleration.

The amount of work done by the external torque is given below:

[Tex]W =\int_{\theta _{0}}^{\theta _{1}}\tau_{ext}(\theta)d\theta = \int_{\theta _{0}}^{\theta _{1}}pESin(\theta)d\theta[/Tex]

W = pE(cos θ₀ – cos θ₁)

As we know that the work done in bringing a system of charges from infinity to the given configuration is defined as the potential energy of the system, hence the potential energy U(θ)  can be associated with the inclination θ of the dipole using the above relation.

Therefore, U(θ) = pE(cos θ₀-cos θ₁)

In case the dipole moves by 90°, the potential energy is given as

U(θ) = pE (cos π/2 – cos θ) = -pE cos θ = -p.E

Significance of Dipole in External Field

The significance of a dipole in an external electric field lies in its ability to interact with the field and experience a torque and potential energy that depend on its orientation relative to the field.

• When placed in an external electric field, a dipole tends to align itself with the field. If the dipole moment vector is parallel to the electric field, the system is at a stable equilibrium, with minimum potential energy.

• The dipole experiences a torque in the presence of an external electric field, which tends to rotate the dipole to align it with the field.

• The dipole also possesses potential energy when placed in an external electric field. This potential energy depends on the orientation of the dipole relative to the field and is minimized when the dipole aligns with the field.

• The behavior of dipoles in external fields is used in in determining molecular structure, reactivity, and solvation.

Conclusion

Here we learned about the dipole in a uniform external field. It consists of two opposed and equal charges, infinitesimally close together, even though the dipoles have different charges. In areas such as physics, we frequently discover that the dimensions of a item may be ignored and represented as a point like object that is a point particle.

Related Reads

• Torque on an Electric Dipole in Uniform Electric Field
• Potential Energy in an External Field
• Dipole in a Uniform Magnetic Field

FAQs on Dipole in a Uniform External Field

What are the real-life applications of a dipole in a uniform external field?

The behavior of dipoles in uniform external fields has applications in various fields such as physics, chemistry, and engineering. Examples include the alignment of molecules in electric or magnetic fields, the operation of devices like electric motors and generators, and the behavior of particles in particle accelerators .

Where is the dipole in unstable equilibrium in a uniform external field?

The dipole is in unstable equilibrium when its dipole moment is perpendicular to the external field. In this orientation, the torque on the dipole is maximum, and any small displacement from equilibrium results in a torque that further rotates the dipole away from its original position.

What is the potential energy of a dipole in a uniform external field?

The potential energy U of a dipole in a uniform external field is given by the dot product of the dipole moment p​ and the external field U = −p​⋅E. This represents the work done by the external field in reorienting the dipole.

How does the orientation of the dipole affect the torque?

The torque experienced by the dipole depends on the angle between the dipole moment and the external field. It is maximum when the dipole moment is perpendicular to the external field and minimum (zero) when the dipole moment is parallel / antiparallel to the field.