# Torque on an Electric Dipole in Uniform Electric Field

Science is a strange subject that never ceases to amaze you as new subjects are presented. We’re all aware that charge occurs everywhere around us and that its existence causes a variety of natural events. Furthermore, positive and negative charges exist in many forms, displaying various characteristics in the presence of a stimulating field.

Have you ever come across the term “electric dipole”? This unusual configuration of electric charges, i.e., positive and negative charges, creates an intriguing physics idea. To be more specific, Electric Dipole is a separation of positive and negative charges.

Consider a pair of electric charges with opposite signs but equal magnitude that are separated by a much smaller distance. The behavior of an Electric Dipole in the presence of an external field is now our main focus. Let’s review the characteristics of the torque acting on an electric dipole in a uniform electric field before moving on to the properties of the torque acting on an electric dipole in a uniform electric field.

### Torque

Torque is the measurement of the force that causes an item to rotate around an axis.

Torque is a vector quantity whose direction is determined by the force acting on the axis. The torque vector’s magnitude is determined as follows:

**τ = F r sinθ**

where

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

### Electric Dipole

An electric dipole is a pair of equal and opposite electric charges separated by a distance d.

The product of the magnitude of these charges and the distance between them is the electric dipole moment. The electric dipole moment is a vector that has a clear direction from negative to positive charge.

The electric dipole moment,

**p = q d**

where

- q is the magnitude of charge &
- d is the separating distance.

### Torque on an electric dipole

Consider a dipole that is placed in a uniform external field ‘E’ to calculate the torque experienced by the dipole when placed in an external field. The positive charge will be subjected to an electric force of ‘qE’ magnitude in the upward direction, while the negative charge will be subjected to an electric force of ‘qE’ magnitude in the downward direction.

Because the net force is zero, it can be observed that the dipole is in transitional equilibrium. What is, however, the rotational equilibrium? In this scenario, the dipole may remain fixed but rotates with a certain angular velocity. This fact has been demonstrated experimentally, and it indicates that both electrostatic forces (qE) behave as clockwise torque. As a result, when the dipole is put in a uniform external electric field, it rotates. Always keep in mind that torque always works in pairs. Furthermore, its magnitude is the resulting product of force and its arm. The arm may be thought of as the distance between the point where force is applied and the point where rotation occurs for the dipole.

**Derivation of Torque**

Consider a dipole with charges +q and –q that form a dipole because they are separated by a distance of d. Place it in a homogeneous electric field of strength E, with the dipole’s axis forming an angle θ with the electric field.

The force on the charges, **F = ± q E**

The components of the force perpendicular to the dipole, **F = ± q E sinθ**

Since these components are equal and are separated by a distance d, the torque on the dipole is:

Torque = Force × distance between forces

τ = (q E sinθ) d = q d E sinθ

Since ‘qd’ is the magnitude of dipole moment (p), and the direction of dipole moment is from positive to negative charge; torque is the cross product of dipole moment and electric field. If the direction of an electric field is positive, the torque is in the clockwise direction (therefore negative) in the above figure.

Thus,

**τ = – p E sinθ**

The negative sign shows that torque is in the clockwise direction.

### Sample Problems

**Problem 1: An electric dipole is placed at an angle of 30° with an electric field of intensity 3 × 10 ^{4} N ⁄ C. It experiences a torque of 5 Nm. Calculate the charge on the dipole if the dipole length is 5 cm.**

**Solution:**

Given:

Electric field, E = 3 × 10

^{4}N ⁄ CAngle between dipole and electric field, θ = 30°

Dipole length, d = 5 cm = 0.05 m

Torque, τ = 5 N m

The torque on a dipole in electric field is given by:

τ = q d E sin θ

q = τ ⁄ d E sin θ

= 5 ⁄ (0.05 × 3 × 10

^{4}× sin30°) C= 6.7 mC

Hence, the torque on a dipole in an electric field is

6.7 mC.

**Problem 2: Two tiny electrical dipoles AB and CD, each with a dipole moment of p, are maintained at an angle of 120**°**. What is the dipole moment as a result? What would the amount and direction of torque operating on this system be if E is directed in the +x direction?**

**Solution:**

Effective dipole moment of AB and CD, p

_{e}= √(p^{2}+ p^{2}+ 2 p p cos120°)= √(2 p

^{2}+ 2 p^{2}(−1 ⁄ 2))= p

The resultant vector makes 30° angle with x-axis, so the torque experienced by the effective dipole,

τ = p E sin 30°

= (1 ⁄ 2) p E

It acts along the z-direction.

Hence, the effective dipole moment is

p; the torque operating on the system is(1 ⁄ 2) p Ealong the+z-direction.

**Problem 3: An electric dipole of magnitude 0.5 C m is placed parallel to an electric field of intensity 30 N⁄C. Calculate the torque acting on the dipole.**

**Solution:**

Given:

Electric field, E = 30 N⁄C

Angle between dipole and electric field, θ = 0°

Electric dipole moment, p = 0.5 C m

The torque acting on a dipole is given as:

τ = p E sinθ

= p E sin0°

= 0

Hence, the torque acting on the dipole is

0.

**Problem 4: Why do a dipole experience force and torque both when placed in a non-uniform electric field?**

**Solution:**

Each charge of a dipole receives a force when it is put in an uniform electric field and the dipole vector direction is not parallel to the field direction. The magnitude of both forces is identical, yet they are moving in opposing directions. A pair is formed by these equal and opposing parallel forces. This pair applies a torque on the dipole, causing it to spin and align in the field direction. The force, however, is always zero in uniform field.

There will be a torque when a dipole is put in a non-uniform field, as stated above. The forces acting on the charges are not the same after the dipole is aligned to the field direction. As a result, the dipole will be subjected to a net force in the direction of increasing field. As a result, the electric dipole experiences both torque and force in a non-uniform field.

**Problem 5: At what angle does the dipole in an external electric field experience the maximum torque on it?**

**Solution:**

The torque acting on a dipole is given as:

τ = p E sinθThe value of sine function is maximum at an angle of 90°, i.e., sin90° = 1, so the dipole experiences the maximum torque at an angle of 90° with the field, or we can say that the dipole experiences the maximum torque when it is held perpendicular to the electric field.

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