A cylindrical capacitor has length L, inner radius A, outer radius B. What is its capacitance?

A cylindrical capacitor has length L, inner radius A, outer radius B. What is its capacitance?

• A. C = (2π ε0 L) / ln(B/A)
• B. C = (π ε0 L) / ln(B/A)
• C. C = (ε0 L) / ln(B/A)
• D. C = (2π ε0 L) ln(B/A)

Answer: (A)

This question tests how the capacitance of a coaxial cylindrical capacitor depends on its geometry. Between the inner radius a and outer radius b, the electric field for a line charge λ on the inner conductor is E(r) = λ/(2π ε0 r). The potential difference between the cylinders is V = ∫ from a to b E·dr = (λ/(2π ε0)) ln(b/a). Therefore the capacitance per unit length is C' = λ/V = (2π ε0)/ln(b/a). For a cylinder of length L, the total capacitance is C = C' L = (2π ε0 L)/ln(B/A). This matches the expression with 2π ε0 L in the numerator and the natural log of the radius ratio in the denominator.

This question tests how the capacitance of a coaxial cylindrical capacitor depends on its geometry. Between the inner radius a and outer radius b, the electric field for a line charge λ on the inner conductor is E(r) = λ/(2π ε0 r). The potential difference between the cylinders is V = ∫ from a to b E·dr = (λ/(2π ε0)) ln(b/a). Therefore the capacitance per unit length is C' = λ/V = (2π ε0)/ln(b/a). For a cylinder of length L, the total capacitance is C = C' L = (2π ε0 L)/ln(B/A). This matches the expression with 2π ε0 L in the numerator and the natural log of the radius ratio in the denominator.