📘 Chapter 23: Electric Potential Energy of an Electric Dipole in an Electrostatic Field (Class XII)

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🔷 1. Introduction

When an electric dipole is placed in an electrostatic field, it possesses potential energy due to its orientation in the field.

This energy determines how the dipole behaves—whether it aligns with the field or resists alignment.

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🔷 2. Concept of Dipole in Electric Field

A dipole consists of:

• Charges +q and −q
• Separation distance 2a
• Dipole moment: p = q × 2a

Placed in a uniform electric field E at angle θ.

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🔷 3. Expression for Potential Energy

The potential energy of a dipole in an electric field is:

U = − pE cosθ

Where:

• U = potential energy
• p = dipole moment
• E = electric field
• θ = angle between p and E

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📦 4. Important Results (Must Remember)

• Potential energy: U = −pE cosθ
• Minimum energy: θ = 0° → U = −pE
• Maximum energy: θ = 180° → U = +pE
• Zero energy: θ = 90°
• Stable equilibrium: θ = 0°
• Unstable equilibrium: θ = 180°

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🔷 5. Variation with Angle

• Energy depends on cosθ
• Changes with orientation
• Independent of position (in uniform field)

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🔷 6. Physical Interpretation

The dipole tends to align itself in the direction of the electric field to minimize its potential energy.

System always moves toward minimum energy configuration

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🔷 7. Relation with Torque

Torque on dipole:

τ = pE sinθ

Relation between torque and potential energy:

τ = − dU/dθ

This shows how energy changes with orientation.

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🔷 8. Energy Diagram

• Minimum at θ = 0°
• Maximum at θ = 180°
• Zero crossing at θ = 90°

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🧠 9. Solved Numerical Problems

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🔹 Q1

Find potential energy of dipole (p = 2×10⁻⁸ C·m) in field 10⁵ N/C at 60°.

Solution:

U = −pE cosθ

= −2×10⁻⁸ × 10⁵ × cos60°

= −2×10⁻³ × 0.5

U = −1×10⁻³ J

Answer: −1×10⁻³ Joule

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🔹 Q2

At what angle is energy minimum?

Answer:

θ = 0°

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🔹 Q3

When is energy zero?

Answer:

θ = 90°

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🔹 Q4

What happens when dipole is opposite to field?

Answer:

Energy is maximum and system is unstable.

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🔹 Q5

Why does dipole align with field?

Answer:

To minimize its potential energy.

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🔷 10. Advanced Conceptual Insight

This concept is fundamental in understanding polar molecules, dielectric materials, and molecular alignment in external fields.

It is also analogous to magnetic dipole behavior in magnetic fields.

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🔷 11. Applications

• Dielectric polarization
• Molecular alignment
• Electric field sensors
• Capacitor behavior

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🔷 12. Summary

The potential energy of a dipole in an electric field depends on its orientation. It is minimum when aligned with the field and maximum when opposite, guiding the dipole’s natural behavior.

✨ End of Chapter 23: Dipole Potential Energy ✨