Equipotential Surfaces: Definition, properties, and diagrams for isolated charges and dipoles

📘 1. Electrostatic Potential Energy

Electrostatic potential energy is the energy stored in a system of charges because of their relative positions.

U = kq₁q₂/r

where k = 1/(4πε₀). Potential energy is generally taken as zero when the charges are infinitely separated.

➕ Like Charges

q₁q₂ > 0

U > 0

➖ Unlike Charges

q₁q₂ < 0

U < 0

∞ Infinite Separation

U = 0

📐 2. Potential Energy of a System of Point Charges

For many point charges, the total electrostatic potential energy is the sum of the interaction energy of every distinct pair.

U = k Σ(qᵢqⱼ/rᵢⱼ),   i < j

For three charges:

U = k[ q₁q₂/r₁₂ + q₂q₃/r₂₃ + q₃q₁/r₃₁ ]

⭐ JEE Point: Self-interaction of a charge is not included. Only distinct pairs are considered.

🔬 3. Interactive Practical — Dipole in Uniform Electric Field

Angle θ 30°

Dipole Moment p 5

Electric Field E 5

Animation

+q

−q

θ =
Potential Energy U = −pE cosθ =
Torque τ = pE sinθ =
Status:

⚡ 4. Potential Energy of an Electric Dipole

An electric dipole consists of equal and opposite charges +q and −q separated by distance 2a.

p = q(2a)

U = −pE cosθ

θ = 0°

Dipole parallel to electric field.

U = −pE

Minimum Energy → Stable

θ = 90°

Dipole perpendicular to electric field.

U = 0

θ = 180°

Dipole antiparallel to electric field.

U = +pE

Maximum Energy → Unstable

🔄 5. Torque and Potential Energy

τ = pE sinθ

τ = −dU/dθ

The torque tends to rotate the dipole toward its minimum potential energy position.

🧮 6. Solved Numerical

Question:
Two charges +2 μC and +3 μC are separated by 0.3 m. Find their electrostatic potential energy.

U = kq₁q₂/r

U = (9×10⁹)(2×10⁻⁶)(3×10⁻⁶)/0.3

U = +0.18 J

📝 7. MCQ Practice

1. Potential energy of two point charges is:

A. kq₁q₂r

B. kq₁q₂/r

C. kq₁q₂/r²

D. kr/q₁q₂

✔ Answer: B

2. Potential energy of two like charges is:

A. Positive

B. Negative

C. Zero

D. Infinite

✔ Answer: A

3. Potential energy of two unlike charges is:

A. Positive

B. Negative

C. Zero

D. Infinite

✔ Answer: B

4. Potential energy of a dipole in a uniform field is:

A. pE sinθ

B. pE cosθ

C. −pE cosθ

D. −pE sinθ

✔ Answer: C

5. Dipole potential energy is minimum at:

A. 0°

B. 45°

C. 90°

D. 180°

✔ Answer: A

6. Dipole potential energy at 90° is:

A. pE

B. −pE

C. 0

D. 2pE

✔ Answer: C

7. Torque on a dipole is:

A. pE cosθ

B. pE sinθ

C. p/E

D. E/p

✔ Answer: B

8. Stable equilibrium corresponds to:

A. Maximum U

B. Minimum U

C. Infinite U

D. Zero charge

✔ Answer: B

9. Potential energy at infinite separation is:

A. +1 J

B. −1 J

C. 0

D. ∞

✔ Answer: C

10. If separation between two charges doubles, U becomes:

A. 2U

B. U/2

C. 4U

D. U/4

✔ Answer: B

🟢 8. 2 Marks — 6 Questions

Q1. Define electrostatic potential energy.

Q2. Write the expression for potential energy of two point charges.

Q3. Why is potential energy of like charges positive?

Q4. Write the potential energy of an electric dipole in a uniform field.

Q5. What is the potential energy of a dipole at θ = 90°?

Q6. What is stable equilibrium of an electric dipole?

🟡 9. 3 Marks — 6 Questions

Q1. Derive U = kq₁q₂/r.

Q2. Explain positive and negative electrostatic potential energy.

Q3. Explain potential energy of three point charges.

Q4. Derive U = −pE cosθ.

Q5. Explain stable and unstable equilibrium of a dipole.

Q6. Write the relation between torque and potential energy.

🟠 10. 4 Marks — 6 Questions

Q1. Derive the potential energy of two point charges.

Q2. Explain potential energy of three point charges.

Q3. Derive the potential energy of a dipole in a uniform field.

Q4. Explain variation of dipole potential energy with θ.

Q5. Explain stable and unstable equilibrium using potential energy.

Q6. Explain the assembly method for finding electrostatic potential energy.

🔴 11. 5 Marks — 6 Questions

Q1. Derive electrostatic potential energy of a system of three point charges.

Q2. Explain the work-energy interpretation of electrostatic potential energy.

Q3. Derive U = −pE cosθ for a dipole.

Q4. Discuss stable and unstable equilibrium of an electric dipole.

Q5. Derive τ = −dU/dθ.

Q6. Explain how potential energy changes with separation between two charges.

🔵 12. 6 Marks — 6 Questions

Q1. Derive the potential energy of a system of three point charges and explain its significance.

Q2. Explain in detail the pairwise interaction method for a system of point charges.

Q3. Derive the potential energy of an electric dipole in a uniform electric field.

Q4. Explain stable and unstable equilibrium of a dipole using potential energy.

Q5. Derive the relation between torque and potential energy of an electric dipole.

Q6. Explain total electrostatic potential energy of a system containing point charges and a dipole.

🚀 Quick Revision

Two Charges: U = kq₁q₂/r

System: U = kΣ(qᵢqⱼ/rᵢⱼ)

Dipole: U = −pE cosθ

Torque: τ = pE sinθ

Stable: θ = 0°, U = −pE

Unstable: θ = 180°, U = +pE