Torque on a Dipole in a Uniform Electric Field

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Introduction

As we are familiar with the concept of electric dipole, let us now look at the concept of torque on a dipole which is a simple topic and a scoring one. A dipole is a magnetised pole with an equal amount of positive and negative charges separated by distance (d). When a dipole is in a uniform electric field, it will experience some form of force and acquire a rotating effect. This rotating effect is known as ‘torque’. The term “torque” comes from Latin, meaning “to twist”. Torque is a vector quantity, and its direction generally depends on the force applied to an object at any point. The Torque can be calculated by estimating the negative and positive charges’ overall rotation in an electric field. To understand the torque on a dipole in a uniform electric field. Let’s first revise the basic terms like “Torque”, “Dipole”, and “electric field”.

Torque

By definition, Torque measures the force that causes an object to rotate about its axis. It is also considered a rotational force, moment, a moment of force, or turning effect. It depends on the direction and magnitude in which one applies the turning momentum. Torque can be thought of as a rotational equivalent to force, and the SI unit of torque is called a newton metre (Nm).
The symbol for torque is typical “τ”. The magnitude of torque depends on 3 variables: the force applied, the length of the lever arm connecting the origin to the point where the force is applied, and the angle between these two.

The torque magnitude is determined as follows:

τ = F r sinθ

F = force acting on the axis,
r = the length of moment arm,
θ = the angle between the force vector and moment arm, &
τ = the torque vector

Dipole

Dipoles are extremely prevalent. They are essentially two charges separated by some type of non-conducting medium (e.g., air, vacuum, etc.).
An example of a dipole can be seen in the electromagnetic wave, where a dipole in a uniform electric field is polarised. A dipole in an electromagnetic system deals with the positive and negative charges separated by a distance. It is characterised by their dipole moment, a vector quantity. 
The forces associated with electric fields are mediated by the movement of charge. In the case of a simple dipole, there will be an attractive force between the positive charge and the negative charge, even if they are very distant from one another.
The dipole moment is p = q x d
q = the magnitude of charge &
d = the separating distance.

Electric Field

The strength of an electric field is measured in Volt/metre [V/(m)]. An electric field has both direction and magnitude, which means it can have different magnitudes, directions, or even both in different regions of space.
The electric field lines are the trajectories of the electric force; whenever an object with a charge moves, through electromagnetic forces, it will follow these paths. The direction that the force is applied on determines which way the field line will be drawn (remember that like charges repel and opposites attract).

The electric field is explained as E = F/q
F = force exerts on the charge by an electric field
q = the charge

Torque on a Dipole in a Uniform Electric Field

Consider an electric dipole placed in a uniform external field ‘E’ to calculate the torque acting on a dipole. It will calculate the torque on a dipole in a uniform electric field. The positive charge experiences a force that is magnitude qE upwards or downwards. At the same time, its opposite charge also experiences this same magnitude but with directions reversed due to polarity between charges (+/-). It can be observed that the net force is zero, and the dipole is in translational equilibrium.
As mentioned above, when torque acts on a dipole, the electric dipole rests in translational equilibrium since the net force is zero. But how does rotational equilibrium come into play here? Considering this case, the dipole might stay in a stationary position.

Derivation of Torque on a Dipole in a Uniform Electric Eield

Consider a dipole with charges +q and –q separated by a distance of d. The axis of the electric dipole forms an angle θ with the electric field when the Electric Dipole is placed in the uniform electric field of strength E.

The force on both the positive and negative charges will be
F+ = + qE

F– = – qE

The forces perpendicular to the dipole is:

F+ ⊥  =  +qEsinθ
F– ⊥  =  -qEsinθ
Since these components are separated by a distance d and are equal, the torque on the dipole is:

T = (q E sinθ) d = q d E sinθ

Here
T = Torque 

q= the charge
E  = Electric field strength 

d= distance between the charges.

Dipole moment is given by:
P = qd (q= charge, d = distance between the charge)

The direction is from positive to negative charge for dipole moments. The torque acting on a dipole is the cross product of electric field and dipole moment as we can look into the formula. 
T = P x E ( Cross product)

The equation has revealed that both electrostatic forces (qE) function as torque when applied in the clockwise direction. Therefore, torque on a dipole in a uniform electric field gets to rotate.

The important thing is to remember that torque always operates in a couple. That’s why its magnitude is equivalent to the sum of the resultant product of force & its arm. The arm can be thought of as the distance between the point at which force operates and the point at which rotation occurs for the dipole in a uniform external field.

Conclusion

The paragraphs above will clear your doubts on the concept of how torque on a dipole in a uniform electric field works. The article discussed the basic terminology of Torque on Dipole in the uniform electric field, including the definition of torque, dipole, and electric field, along with the formulae. The article also discussed derivation and the method of calculating torque. For further understanding and in-depth knowledge, read more articles on the electric field, electric field due to a point charge, electric field lines, and electric dipole.