Two conducting spheres of radii $R_1$ and $R_2$ (where $R_1 > R_2$) are kept at a very large distance from each other. They are charged to the same electrostatic potential and then connected by a fine conducting wire. The ratio of their final surface charge densities ($\sigma_1 / \sigma_2$) will be:

Two conducting spheres of radii R₁ and R₂ (R₁ > R₂) are kept at a very large distance from each other. They are charged to the same electrostatic potential and then connected by a fine conducting wire. The ratio of their final surface charge densities +

Correct Answer

σ₁ / σ₂ = R₂ / R₁

Step-by-Step Solution

For an isolated conducting sphere, the potential is:

V = kQ/R

Initially, both spheres are charged to the same potential. Therefore:

kQ₁/R₁ = kQ₂/R₂

Hence:

Q₁/Q₂ = R₁/R₂

After connecting them with a conducting wire, they must again reach the same final potential. Therefore:

Q₁/R₁ = Q₂/R₂

So the final charges are also proportional to their radii:

Q₁ : Q₂ = R₁ : R₂

Surface Charge Density

Surface charge density is:

σ = Q/(4πR²)

Therefore:

σ₁/σ₂ = [Q₁/(4πR₁²)] / [Q₂/(4πR₂²)]

Using:

Q₁/Q₂ = R₁/R₂

we get:

σ₁/σ₂ = (R₁/R₂) × (R₂²/R₁²)

Therefore:

σ₁/σ₂ = R₂/R₁

Important Concept

Connected conducting spheres → Same potential.

For isolated spheres: V ∝ Q/R, so Q ∝ R when potentials are equal.

But surface area is proportional to R². Therefore:

σ = Q/(4πR²) ∝ 1/R

JEE Quick Trick

For two connected spheres:

V₁ = V₂  →  Q₁/R₁ = Q₂/R₂

σ ∝ 1/R

So the smaller sphere has the greater surface charge density.

30 Related MCQs with Solutions

1. Two conducting spheres are connected by a wire. Their final potentials are:

A. Different

B. Equal

C. Zero

D. Infinite

Answer: B. Equal

2. The potential of an isolated conducting sphere is:

A. kQR

B. kQ/R

C. kR/Q

D. Q/R²

Answer: B. kQ/R

3. For equal potential, charge on a conducting sphere varies as:

A. 1/R

B. R

C. R²

D. R³

Answer: B. R

4. Surface charge density of a sphere is:

A. Q/(4πR)

B. Q/(4πR²)

C. QR²

D. Q/R³

Answer: B. Q/(4πR²)

5. If R₁ > R₂, which sphere has greater final surface charge density?

A. Sphere 1

B. Sphere 2

C. Both equal

D. Cannot be determined

Answer: B. Sphere 2

6. The ratio σ₁/σ₂ is:

A. R₁/R₂

B. R₂/R₁

C. (R₁/R₂)²

D. 1

Answer: B. R₂/R₁

7. If R₁ = 2R₂, then σ₁/σ₂ equals:

A. 2

B. 1/2

C. 4

D. 1/4

Answer: B. 1/2

8. If R₁ = 3R₂, then σ₂/σ₁ is:

A. 1/3

B. 3

C. 9

D. 1

Answer: B. 3

9. When two conducting spheres are connected, charge flows until:

A. Charges become equal

B. Potentials become equal

C. Radii become equal

D. Surface areas become equal

Answer: B. Potentials become equal

10. Which sphere has more total charge when both have equal potential?

A. Smaller sphere

B. Larger sphere

C. Both equal

D. Neither

Answer: B. Larger sphere

11. For equal potential, Q₁/Q₂ is:

A. R₂/R₁

B. R₁/R₂

C. 1

D. R₁²/R₂²

Answer: B. R₁/R₂

12. Surface area of a sphere is proportional to:

A. R

B. R²

C. 1/R

D. R³

Answer: B. R²

13. For equal potential, surface charge density varies as:

A. R

B. R²

C. 1/R

D. R³

Answer: C. 1/R

14. The electric field just outside a conductor is related to surface charge density by:

A. E = σ/ε₀

B. E = ε₀σ

C. E = σR

D. E = R/σ

Answer: A. E = σ/ε₀

15. Therefore, at the surfaces of the two spheres:

A. E₁/E₂ = R₁/R₂

B. E₁/E₂ = R₂/R₁

C. E₁ = E₂ always

D. E = 0

Answer: B. R₂/R₁

16. Which sphere has the stronger electric field immediately outside its surface?

A. Larger sphere

B. Smaller sphere

C. Both equal

D. Neither

Answer: B. Smaller sphere

17. If the two spheres have equal radii, their final surface charge densities are:

A. Equal

B. Unequal

C. Opposite

D. Zero

Answer: A. Equal

18. The conductor wire ensures that both spheres have:

A. Equal charge

B. Equal potential

C. Equal radius

D. Equal surface area

Answer: B. Equal potential

19. If R₁ = 4R₂, then Q₁/Q₂ is:

A. 1/4

B. 4

C. 16

D. 1

Answer: B. 4

20. If R₁ = 4R₂, then σ₁/σ₂ is:

A. 4

B. 1/4

C. 16

D. 1/16

Answer: B. 1/4

21. Capacitance of an isolated spherical conductor is proportional to:

A. R

B. R²

C. 1/R

D. R³

Answer: A. R

22. A larger conducting sphere at the same potential carries:

A. Less charge

B. More charge

C. No charge

D. Same charge always

Answer: B. More charge

23. The relation between charge and radius at equal potential is:

A. Q ∝ R

B. Q ∝ R²

C. Q ∝ 1/R

D. Q ∝ R³

Answer: A. Q ∝ R

24. Which quantity is inversely proportional to radius for equal potential spheres?

A. Total charge

B. Surface charge density

C. Capacitance

D. Surface area

Answer: B. Surface charge density

25. If σ₁ < σ₂, then:

A. R₁ < R₂

B. R₁ > R₂

C. R₁ = R₂

D. No relation

Answer: B. R₁ > R₂

26. At electrostatic equilibrium, the potential throughout a conductor is:

A. Variable

B. Constant

C. Zero always

D. Infinite

Answer: B. Constant

27. When connected spheres are far apart, their mutual influence can be:

A. Neglected approximately

B. Infinite

C. Exactly zero in every case

D. Dominant

Answer: A. Neglected approximately

28. For two connected spheres, which statement is correct?

A. Equal charge density

B. Equal potential

C. Equal charge

D. Equal electric field everywhere

Answer: B. Equal potential

29. The smaller sphere has greater surface charge density because:

A. It carries no charge

B. Its surface area is smaller for its charge

C. Its potential is different

D. Its capacitance is zero

Answer: B

30. The final ratio σ₁/σ₂ for R₁ > R₂ is:

A. R₁/R₂

B. R₂/R₁

C. 1

D. R₁²/R₂²

Answer: B. R₂/R₁

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