Question

Two identical balls having like charges and placed at a certain distance apart repel each other with a certain force. They are brought in contact and then moved apart to a distance equal to half their initial separation. The force of repulsion between them increases 4.5 times in comparison with the initial value. The ratio of the initial charges of the balls is……. 

(A) 4 : 1 

(B) 6 : 1 

(C) 3 : 1 

(D) 2 : 1

Answer 1

The correct option (D) 2 : 1  

Explanation:

Let balls have charges Q₁ & Q₂ respectively

∴ F = [(kQ₁Q₂)/r²] (1)

when two charges are brought in contact & moved apart then they acquire charge [(Q₁ + Q₂)/2]

∴ in final condition, F' = [k{(Q₁ + Q₂)/2}²/(r/2)²]

 ∴ F' = [{k(Q₁ + Q₂)²}/r²]  (2)

as F' = 4.5F hence from (1) & (2)

[{k(Q₁ + Q₂)²}/r²] = 4.5 [(kQ₁Q₂)/r²]

∴ (Q₁ + Q₂)² = 4.5 Q₁Q₂

∴ Q₁² + Q₂² + 2Q₁Q₂ = 4.5 Q₁Q₂

∴ Q₁² + Q₂² – 2.5 Q₁Q₂ = 0

∴ (Q₁/Q₂)² + 1 – 2.5(Q₁Q₂) = 0 ------- dividing by Q₂²

Let (Q₁/Q₂) = x

∴ x² – 2.5x + 1 = 0

x² – 2x – 0.5x + 1 = 0

x(x – 2) – 0.5x(x – 2) = 0

x = 2   or  x = 0.5

∴ (Q₁/Q₂) = (1/2) or (Q₁/Q₂) = 2