Dipole in a Uniform Field: Derivation of torque acting on a dipole and its potential energy

⚡ Dipole in a Uniform Electric Field: Torque & Potential Energy +

1. Dipole Placed in a Uniform Electric Field

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

p = q(2a)

Now place this dipole in a uniform electric field E. Let the angle between the dipole moment and electric field be θ.

🔴 +q experiences force qE in the direction of E.

🔵 −q experiences force qE opposite to E.

The two forces are equal and opposite, so: Net force = 0 But their lines of action are different, producing a torque.

2. ⚡ Practical Animated Dipole Simulator

Change θ, q and E to observe Torque & Potential Energy

Angle θ

θ = 60°

Charge q

q = 5 μC

Dipole Separation 2a

2a = 4 m

Electric Field E

E = 5 N/C

−

+

Uniform Electric Field E →

Dipole Moment p: 0 C·m
Torque τ: 0 N·m
Potential Energy U: 0 J
Stable / Unstable: Neither
Observation: Torque is maximum at 90°.

3. Derivation of Torque Acting on a Dipole

Let the dipole be placed at angle θ with the uniform electric field E. Force on +q:

F = qE

Force on −q:

F = qE

These forces are equal and opposite. Therefore:

Net Force = 0

However, they form a couple. The perpendicular distance between their lines of action is:

2a sin θ

Hence torque of the couple:

τ = Force × perpendicular distance

Therefore:

τ = qE(2a sin θ)

Since:

p = q(2a)

We obtain:

τ = pE sin θ

Vector form:

τ⃗ = p⃗ × E⃗

4. Important Values of Torque

Angle θ	sin θ	Torque
0°	0	0
30°	1/2	pE/2
60°	√3/2	√3pE/2
90°	1	pE maximum
180°	0	0

5. Potential Energy of an Electric Dipole

The torque acting on the dipole is:

τ = pE sin θ

For a small angular displacement:

dW = τ dθ

The work done by the external agent in slowly rotating the dipole gives the increase in potential energy. Thus:

dU = pE sin θ dθ

Integrating:

U = ∫ pE sin θ dθ

Therefore:

U = −pE cos θ + C

Taking the zero of potential energy at θ = 90°:

U = −pE cos θ

6. Potential Energy at Important Positions

Position	θ	U = −pE cos θ	Nature
Parallel	0°	−pE	Stable
Perpendicular	90°	0	Unstable equilibrium point
Antiparallel	180°	+pE	Unstable

7. ⭐ Stable & Unstable Equilibrium

θ = 0°

Dipole moment is parallel to E.

Torque = 0

Potential energy is minimum:
U = −pE

✅ Stable equilibrium

θ = 180°

Dipole moment is opposite to E.

Torque = 0

Potential energy is maximum:
U = +pE

❌ Unstable equilibrium

θ = 90°

Torque is maximum:
τ = pE

Potential energy:
U = 0

Dipole tends to rotate toward θ = 0°.

8. Work Done in Rotating a Dipole

Work done by external agent in rotating the dipole from θ₁ to θ₂:

W_external = pE(cos θ₁ − cos θ₂)

Change in potential energy:

ΔU = U₂ − U₁

Therefore:

ΔU = pE(cos θ₁ − cos θ₂)

Work done by electric field:

W_field = −ΔU

9. Energy Range

Since:

U = −pE cos θ

and:

−1 ≤ cos θ ≤ 1

Therefore:

−pE ≤ U ≤ +pE

Hence:

Minimum potential energy = −pE

Maximum potential energy = +pE

Total range of potential energy = 2pE

10. 🔬 Practical Interpretation

Think of the electric dipole like a tiny compass. The uniform electric field tries to rotate the dipole so that its dipole moment p becomes parallel to the electric field E.

When: θ = 0° → dipole is aligned → stable.

When: θ = 90° → turning effect is maximum.

When: θ = 180° → dipole is completely reversed → unstable.

11. ⭐ JEE Quick Formula Sheet

Dipole moment: p = q × 2a

Torque: τ = pE sin θ

Vector torque: τ⃗ = p⃗ × E⃗

Potential energy: U = −pE cos θ

Maximum torque: τ_max = pE

Minimum energy: U_min = −pE

Maximum energy: U_max = +pE

Energy range: −pE ≤ U ≤ +pE

12. 📝 30 MCQs with Solutions

1. An electric dipole is placed in a uniform electric field. The net force on the dipole is:

A. qE

B. 2qE

C. Zero

D. pE

Answer: C. Zero
The two forces qE are equal and opposite.

2. Torque acting on a dipole in a uniform electric field is:

A. pE cos θ

B. pE sin θ

C. p/E

D. E/p

Answer: B. pE sin θ

3. Torque is maximum when θ is:

A. 0°

B. 30°

C. 90°

D. 180°

Answer: C. 90°
sin 90° = 1.

4. Torque is zero when θ is:

A. 0° only

B. 90° only

C. 180° only

D. 0° and 180°

Answer: D. 0° and 180°

5. Vector form of torque is:

A. p·E

B. p×E

C. E×p with same convention

D. p/E

Answer: B. τ⃗ = p⃗ × E⃗

6. Potential energy of a dipole is:

A. pE cos θ

B. −pE cos θ

C. pE sin θ

D. −pE sin θ

Answer: B. −pE cos θ

7. Potential energy is minimum when:

A. θ = 0°

B. θ = 45°

C. θ = 90°

D. θ = 180°

Answer: A. θ = 0°
U = −pE.

8. Potential energy is maximum when:

A. θ = 0°

B. θ = 60°

C. θ = 90°

D. θ = 180°

Answer: D. θ = 180°

9. Stable equilibrium corresponds to:

A. θ = 0°

B. θ = 90°

C. θ = 180°

D. θ = 270°

Answer: A. θ = 0°

10. Unstable equilibrium corresponds to:

A. θ = 0°

B. θ = 45°

C. θ = 90°

D. θ = 180°

Answer: D. θ = 180°

11. If p is doubled, torque at fixed E and θ becomes:

A. p/2

B. Same

C. 2 times

D. 4 times

Answer: C. 2 times

12. If electric field is doubled, torque becomes:

A. Half

B. Same

C. Double

D. Four times

Answer: C. Double

13. At θ = 90°, potential energy is:

A. pE

B. −pE

C. Zero

D. 2pE

Answer: C. Zero

14. Maximum torque is:

A. p/E

B. pE

C. E/p

D. p²E

Answer: B. pE

15. The SI unit of torque is:

A. N/C

B. C·m

C. N·m

D. J/C

Answer: C. N·m

16. The SI unit of dipole moment is:

A. C·m

B. N·m

C. N/C

D. C/N

Answer: A. C·m

17. The dipole tends to align itself:

A. Perpendicular to E

B. Parallel to E

C. Opposite to E

D. Randomly

Answer: B. Parallel to E

18. Work done by the electric field is related to potential energy by:

A. W = ΔU

B. W = −ΔU

C. W = 2ΔU

D. W = 0 always

Answer: B. W_field = −ΔU

19. If θ changes from 0° to 90°, potential energy:

A. Increases from −pE to 0

B. Decreases from 0 to −pE

C. Remains same

D. Becomes +pE

Answer: A

20. If θ changes from 90° to 180°, potential energy:

A. Decreases

B. Increases from 0 to +pE

C. Becomes −pE

D. Remains zero

Answer: B

21. For θ = 30°, torque is:

A. pE

B. pE/2

C. √3pE/2

D. Zero

Answer: B. pE/2

22. For θ = 60°, torque is:

A. pE/2

B. √3pE/2

C. pE

D. Zero

Answer: B. √3pE/2

23. The total range of potential energy of a dipole is:

A. pE

B. 2pE

C. pE/2

D. Zero

Answer: B. 2pE

24. At θ = 180°, torque is:

A. pE

B. −pE

C. Zero

D. 2pE

Answer: C. Zero

25. At θ = 0°, torque is:

A. Maximum

B. Zero

C. pE

D. 2pE

Answer: B. Zero

26. A dipole in a uniform electric field experiences:

A. Net force only

B. Torque only in general

C. Neither force nor torque

D. Infinite force

Answer: B
Net force is zero, while torque can be non-zero.

27. If p = 2 C·m, E = 5 N/C and θ = 90°, torque is:

A. 2 N·m

B. 5 N·m

C. 10 N·m

D. 20 N·m

Answer: C. 10 N·m
τ = pE sin90° = 2 × 5 = 10 N·m.

28. If p = 2 C·m, E = 5 N/C and θ = 0°, potential energy is:

A. +10 J

B. −10 J

C. 0

D. 20 J

Answer: B. −10 J
U = −pE = −2 × 5 = −10 J.

29. If p = 2 C·m, E = 5 N/C and θ = 180°, potential energy is:

A. −10 J

B. 0

C. +10 J

D. +5 J

Answer: C. +10 J

30. The torque vector on a dipole is perpendicular to:

A. Only p

B. Only E

C. Both p and E

D. Neither

Answer: C. Both p and E
Because τ⃗ = p⃗ × E⃗.

13. ⭐ Final Revision

Net Force: F_net = 0

Torque: τ = pE sinθ

Vector Torque: τ⃗ = p⃗ × E⃗

Maximum Torque: τ_max = pE at 90°

Potential Energy: U = −pE cosθ

Stable: θ = 0°, U = −pE

Unstable: θ = 180°, U = +pE

Energy Range: −pE ≤ U ≤ +pE