Potential Energy in an External Field - Potential Energy of a Dipole in an External Field

When an electric dipole is placed in a uniform electric field, the net force on it is zero, but a torque acts on it if the dipole is inclined to the field. This torque tries to rotate the dipole so that it aligns with the field direction.

If the dipole rotates under the influence of the field, work is involved. That work gets stored as potential energy depending on the angle between the dipole moment p and the electric field E.

• In a uniform field, the net force is zero on a dipole.
• In a uniform field, torque may be non-zero if the dipole is not parallel to the field.
• The dipole tends to move toward the position of minimum potential energy.

For a dipole making an angle θ with a uniform electric field:

τ = pE sin θ

In vector form: τ = p × E

This torque rotates the dipole toward the field direction.

Suppose the dipole is rotated slowly from angle θ₀ to θ against the electric field. An external agent must do work equal to the increase in potential energy.

Step 1: External torque

• To rotate the dipole slowly, the external torque must balance the field torque in magnitude.
• τ_ext = pE sin ⁡θ

Step 2: Small work done

• For a small angular displacement dθ, the work done is:
  dW = τ_ext dθ = pE sin⁡θ dθ

Step 3: Total work done

• Integrating from θ₀ to θ:
  W = \[\int_{\theta_0}^\theta pE\sin\theta d\theta=pE(\cos\theta_0-\cos\theta)\]
• This work is stored as a potential energy difference.

Step 4: Choosing reference position

• If potential energy is taken as zero at θ₀ = 90°, then cos 90° = 0.
• So, U(θ) = −pE cos ⁡θ

Final Formula

Example

A molecule has a dipole moment of 10^-29 C m. One mole of such molecules is placed in an electric field of strength 10⁶ V m⁻¹. If the direction of the electric field is suddenly changed by 60°, find the change in potential energy.

Given

• Dipole moment of one molecule = 10^-29 C m.
• Number of molecules in one mole = 6 × 10²³.
• Electric field = 10⁶ V m⁻¹.
• Angular change = 60°.

Step 1: Total dipole moment

Step 2: Initial potential energy

• Assume initially that dipoles are aligned with the field.
• U_i = −pE = −(6 × 10⁻⁶)(10⁶) = −6 J

Step 3: Final potential energy

• After field changes by 60°: U_f = −pE cos ⁡60^∘ = −(6)(1/2) = −3 J

Answer

The potential energy increases by 3 J. This energy is associated with the new orientation relative to the changed field direction.