Two isolated conducting spheres of radii R and 2R are initially charged such that both have the same surface charge density ₀ . They are then connected by a long, thin conducting wire. After electrostatic equilibrium is reached, the new surface charge density of the smaller sphere will be

Step-by-step solution

The initial charges on the spheres are: Q₁ = ₀ (4 R^2) Q₂ = ₀ (4 (2R)^2) = 4 ₀ (4 R^2) = 4Q₁ By conservation of charge, the total charge is: Q_ total = Q₁ + Q₂ = 5Q₁ When connected by a conducting wire, their potentials become equal ( V₁ = V₂ ): k q₁ R = k q₂ 2R q₂ = 2q₁ Using charge conservation again: q₁ + q₂ = Q_ total q₁ + 2q₁ = 5Q₁ 3q₁ = 5Q₁ q₁ = 5 3 Q₁ The new surface charge density of the smaller sphere is: ₁ = q₁ 4 R^2 = 5 3 ₀ (4 R^2) 4 R^2 = 5 3 ₀ Answer: 5 3 ₀