📘 Chapter 20: Electric Potential Due to a Dipole and System of Charges (Class XII)

🔷 1. Introduction

In previous chapters, we studied electric potential due to a single charge. Now, we extend this concept to more complex systems such as electric dipoles and systems of multiple charges.

These concepts are essential for understanding molecular structures, polarization, and advanced electrostatics.

🔷 2. Electric Potential Due to a Dipole

Consider an electric dipole consisting of charges +q and −q separated by distance 2a.

At a point P located at distance r making angle θ with dipole axis:

V = (1 / 4πε₀) (p cosθ / r²)

Where:

  • p = q × 2a (dipole moment)
  • θ = angle between dipole axis and position vector

📦 3. Important Result (Dipole Potential)

V = (1 / 4πε₀) (p cosθ / r²)

Valid for r ≫ a

🔷 4. Special Cases of Dipole Potential

  • Axial point (θ = 0°): V = (1/4πε₀)(p / r²)
  • Equatorial point (θ = 90°): V = 0

Thus, potential is zero along equatorial line.

🔷 5. Nature of Dipole Potential

  • Depends on both distance and orientation
  • Varies as 1/r²
  • Can be positive or negative

🔷 6. Electric Potential Due to System of Charges

For multiple charges, potential is obtained using principle of superposition.

V = V₁ + V₂ + V₃ + ...

For n charges:

V = (1 / 4πε₀) (q₁/r₁ + q₂/r₂ + q₃/r₃ + ...)

📦 7. Important Results (System of Charges)

  • Potential is scalar → algebraic sum
  • No vector addition required
  • Independent of path
  • Works for any configuration

🔷 8. Physical Interpretation

Electric potential due to a system represents the combined energy effect of all charges at a point.

Dipole potential shows how opposite charges create directional dependence.

🔷 9. Equipotential Surfaces

  • For dipole → complex shapes
  • For multiple charges → irregular surfaces

Always perpendicular to electric field.

🧠 10. Solved Numerical Problems

🔹 Q1

Find potential due to dipole of moment 3×10⁻⁸ C·m at 1 m on axial line.

Solution:

V = (1/4πε₀)(p/r²)

= 9×10⁹ × 3×10⁻⁸ / 1²

V = 270 V

Answer: 270 Volt

🔹 Q2

What is potential at equatorial line?

Answer:

Zero

🔹 Q3

Find potential due to charges 2C and −1C at same distance 1 m.

Solution:

V = k(2/1 − 1/1) = k(1)

V = 9×10⁹ V

Answer: 9×10⁹ Volt

🔹 Q4

Why is potential scalar?

Answer:

Because it depends on work, which is scalar.

🔹 Q5

How does dipole potential vary with distance?

Answer:

As 1/r²

🔷 11. Advanced Conceptual Insight

Dipole potential is fundamental in understanding molecular polarity, dielectric materials, and electromagnetic theory.

Superposition principle allows solving complex real-world problems involving multiple charges.

🔷 12. Applications

  • Molecular physics
  • Dielectric materials
  • Electrostatic energy calculations
  • Charge distribution analysis

🔷 13. Summary

Electric potential due to a dipole depends on both distance and orientation, while potential due to multiple charges is obtained by simple algebraic addition. These concepts are essential for advanced electrostatics.

✨ End of Chapter 20: Dipole and System of Charges ✨