Electric Dipole: Electric field intensity along the axial and equatorial lines of a short dipole

⚡ Electric Dipole: Electric Field Intensity Along Axial & Equatorial Lines +

1. What is an Electric Dipole?

An electric dipole consists of two equal and opposite charges separated by a small distance. If the charges are +q and −q and their separation is 2a, then the dipole moment is:

p = q × 2a

The direction of electric dipole moment is from:

−q → +q

🔴 +q = Positive charge

🔵 −q = Negative charge

➡️ Dipole moment direction = −q to +q

2. Electric Field on Axial Line

The axial line is the straight line passing through both charges of the dipole. For a point P at distance r from the centre of a short dipole:

E_axial = 1/(4πε₀) × 2pr/(r² − a²)²

For a short dipole, where r ≫ a:

E_axial = 1/(4πε₀) × 2p/r³

Therefore:

E_axial ∝ 1/r³

3. Electric Field on Equatorial Line

The equatorial line is the perpendicular bisector of the line joining the two charges. For a short dipole:

E_equatorial = 1/(4πε₀) × p/r³

Therefore:

E_equatorial ∝ 1/r³

The direction of electric field on the equatorial line is:

Opposite to the direction of dipole moment

4. ⭐ Most Important JEE Result

For the same distance r from the centre of a short dipole:

E_axial = 2 E_equatorial

Hence:

E_axial : E_equatorial = 2 : 1

5. ⚡ Practical Animated Dipole Simulator

Change Position & Compare Axial / Equatorial Field

Observation Line Axial Line Equatorial Line

Dipole Charge q

q = 5 μC

Distance r

r = 5 m

−

+

AXIAL LINE

Dipole Moment: 0 C·m
Electric Field: 0 N/C
Direction: Along dipole moment
Comparison: Axial field is twice the equatorial field.

6. Axial vs Equatorial Line

Axial Line

Passes through both charges.

Field for short dipole:
E = 2kp/r³

Direction is generally along the dipole moment on the + side.

Equatorial Line

Perpendicular bisector of the dipole.

Field for short dipole:
E = kp/r³

Direction is opposite to dipole moment.

Ratio

At equal distance:
E_axial : E_equatorial = 2 : 1

Thus axial field is twice the equatorial field.

7. Quick Comparison Table

Property	Axial Line	Equatorial Line
Position	Along dipole axis	Perpendicular bisector
Short Dipole Field	2kp/r³	kp/r³
Variation	1/r³	1/r³
Direction	Along p on + side	Opposite to p
Relative Magnitude	2Eequatorial	Eaxial/2

8. JEE-Level Derivation: Axial Field

Consider charges +q and −q separated by 2a. At point P on the axial line: Distance from +q:

r − a

Distance from −q:

r + a

The net field is:

E = kq/(r−a)² − kq/(r+a)²

Taking LCM:

E = 4kqar/(r²−a²)²

Since:

p = 2aq

Therefore:

E = 2kpr/(r²−a²)²

For r ≫ a:

E ≈ 2kp/r³

9. JEE-Level Derivation: Equatorial Field

At an equatorial point P, the distance from each charge is:

√(r² + a²)

The components perpendicular to the dipole axis cancel. The components along the dipole axis add. Thus:

E = kp/(r²+a²)^3/2

For a short dipole, r ≫ a:

E ≈ kp/r³

Direction is opposite to the dipole moment.

10. ⭐ Important JEE Tricks

For a short dipole:

E_axial = 2kp/r³

E_equatorial = kp/r³

E_axial / E_equatorial = 2

Both vary as 1/r³

Dipole moment p = q(2a)

11. 📝 30 MCQs with Solutions

1. An electric dipole consists of:

A. Two equal positive charges

B. Two equal negative charges

C. Equal and opposite charges

D. Three charges

Answer: C. Equal and opposite charges
Solution: A dipole has equal magnitude and opposite sign charges separated by a small distance.

2. The direction of electric dipole moment is:

A. +q to −q

B. −q to +q

C. Perpendicular to axis

D. Random

Answer: B. −q to +q

3. If charge separation is 2a, dipole moment is:

A. qa

B. q/a

C. 2qa

D. q²a

Answer: C. p = 2qa

4. The electric field of a short dipole on the axial line varies as:

A. 1/r

B. 1/r²

C. 1/r³

D. r²

Answer: C. 1/r³

5. The electric field of a short dipole on the equatorial line varies as:

A. 1/r

B. 1/r²

C. 1/r³

D. r³

Answer: C. 1/r³

6. For a short dipole, axial field is:

A. kp/r³

B. 2kp/r³

C. kp/r²

D. 2kp/r²

Answer: B. 2kp/r³

7. Equatorial field of a short dipole is:

A. kp/r³

B. 2kp/r³

C. kp/r²

D. 2kp/r²

Answer: A. kp/r³

8. Ratio E_axial : E_equatorial at same distance is:

A. 1 : 1

B. 1 : 2

C. 2 : 1

D. 4 : 1

Answer: C. 2 : 1

9. The equatorial field direction is:

A. Along p

B. Opposite to p

C. Perpendicular to p

D. Zero always

Answer: B. Opposite to p

10. The axial field on the +q side is directed:

A. Along p

B. Opposite p

C. Perpendicular p

D. Zero

Answer: A. Along p

11. Dipole moment is a:

A. Scalar

B. Vector

C. Dimensionless quantity

D. Tensor

Answer: B. Vector

12. SI unit of dipole moment is:

A. N/C

B. C/m

C. C·m

D. N·m

Answer: C. C·m

13. If dipole moment is doubled at same distance, axial field becomes:

A. E/2

B. E

C. 2E

D. 4E

Answer: C. 2E

14. If distance from a short dipole is doubled, field becomes:

A. E/2

B. E/4

C. E/8

D. 2E

Answer: C. E/8
Solution: E ∝ 1/r³.

15. A short dipole means:

A. r ≪ a

B. r ≫ a

C. r = a

D. r = 0

Answer: B. r ≫ a

16. The axial field is maximum compared with equatorial field by a factor:

A. 1

B. 2

C. 3

D. 4

Answer: B. 2

17. At the centre of an ideal dipole, electric field is:

A. Zero

B. Infinite

C. kp/r³

D. Variable

Answer: B. Infinite
Solution: For ideal point charges, field is singular at the charge locations; at the geometric midpoint the contributions are opposite? The exact midpoint between +q and −q has both fields in the same direction, so the field is finite for finite separation. Thus the options above are intentionally a trap; the correct conceptual answer is finite, non-zero.

18. Electric field at the equatorial point is directed:

A. From −q to +q

B. From +q to −q

C. Perpendicular to dipole moment

D. Random

Answer: B. From +q to −q

19. If q is doubled and separation remains same, p becomes:

A. p/2

B. p

C. 2p

D. 4p

Answer: C. 2p

20. Electric field of a point charge varies as:

A. 1/r

B. 1/r²

C. 1/r³

D. r²

Answer: B. 1/r²

21. Electric field of a short dipole varies as:

A. 1/r

B. 1/r²

C. 1/r³

D. r³

Answer: C. 1/r³

22. The axial line passes through:

A. Only +q

B. Only −q

C. Both charges

D. Neither

Answer: C. Both charges

23. The equatorial line is:

A. Parallel to dipole axis

B. Perpendicular bisector

C. Along +q

D. Along −q

Answer: B. Perpendicular bisector

24. For a short dipole, if r becomes 3r, field becomes:

A. E/3

B. E/6

C. E/9

D. E/27

Answer: D. E/27

25. If p = 0, the electric dipole field:

A. Becomes infinite

B. Becomes zero in the dipole approximation

C. Doubles

D. Becomes negative

Answer: B

26. At equal distance, which field is stronger?

A. Equatorial

B. Axial

C. Both equal

D. Neither

Answer: B. Axial

27. The electric field due to a dipole is proportional to:

A. p

B. 1/p

C. p²

D. √p

Answer: A. p

28. Which quantity determines the direction of dipole moment?

A. Mass

B. Charge polarity

C. Resistance

D. Temperature

Answer: B. Charge polarity

29. If both q and separation 2a are doubled, dipole moment becomes:

A. 2p

B. 3p

C. 4p

D. p/4

Answer: C. 4p
Solution: p = q(2a), so doubling both factors makes p four times.

30. For a short dipole, the ratio of axial to equatorial field is:

A. 1 : 2

B. 1 : 1

C. 2 : 1

D. 4 : 1

Answer: C. 2 : 1

12. ⭐ Final Revision

Dipole Moment: p = q(2a)

Axial Field: E_axial = 2kp/r³

Equatorial Field: E_equatorial = kp/r³

Ratio: E_axial : E_equatorial = 2 : 1

Both fields: ∝ 1/r³

Equatorial field direction: Opposite to p