The capacitance of an isolated sphere is increased n times when it is enclosed by an earthed concentric sphere. The ratio of their radii is:
Correct answer: B. Radii ratio is 1:n
- A. Radii ratio is n:1
- B. Radii ratio is 1:n
- C. Radii ratio is √n:1
- D. Radii ratio is 1:√n
Explanation
The capacitance of an isolated sphere is given by C = 4πε₀r, where r is the radius of the sphere. When a sphere is enclosed by an earthed concentric sphere, the capacitance becomes C' = 4πε₀(r₁r₂)/(r₂ - r₁), where r₁ is the radius of the inner sphere and r₂ is the radius of the outer sphere. The problem states that the capacitance increases n times, thus C' = nC. Solving the equation n(C = 4πε₀r) = 4πε₀(r₁r₂)/(r₂ - r₁) leads to the ratio of radii r₁/r₂ = 1/n. Therefore, the ratio of the radii is 1:n. Other options suggest incorrect ratios that do not satisfy the given conditions based on the formula for capacitance.
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