A spherical capacitor has inner radius a and outer radius b. What is its capacitance?

A spherical capacitor has inner radius a and outer radius b. What is its capacitance?

• A. C = 4π ε0 a /(b − a)
• B. C = 4π ε0 b /(b − a)
• C. C = 4π ε0 (a + b) /(b − a)
• D. C = 4π ε0 a b /(b − a)

Answer: (D)

The essential idea is to relate charge, potential, and geometry for a spherical capacitor using symmetry and Gauss’s law. Between the inner radius a and outer radius b, the electric field is radial and has magnitude E(r) = Q/(4π ε0 r^2). To find the potential difference, integrate the field from the inner surface to the outer surface: V = ∫ from a to b of E dr = ∫_a^b Q/(4π ε0 r^2) dr = (Q/(4π ε0))(1/a − 1/b) = Q/(4π ε0) (b − a)/(ab). The capacitance is C = Q/V, which yields C = 4π ε0 ab/(b − a). This result also matches physical limits: as the outer radius goes to infinity, the capacitance approaches that of an isolated sphere, 4π ε0 a; as the two spheres get very close (b → a), the capacitance grows without bound. Alternatives that miss the ab product in the numerator or replace it with a single radius fail to capture how both radii interact in shaping the field and the potential, and they do not reproduce the correct limiting behavior.

The essential idea is to relate charge, potential, and geometry for a spherical capacitor using symmetry and Gauss’s law. Between the inner radius a and outer radius b, the electric field is radial and has magnitude E(r) = Q/(4π ε0 r^2).

To find the potential difference, integrate the field from the inner surface to the outer surface: V = ∫ from a to b of E dr = ∫_a^b Q/(4π ε0 r^2) dr = (Q/(4π ε0))(1/a − 1/b) = Q/(4π ε0) (b − a)/(ab).

The capacitance is C = Q/V, which yields C = 4π ε0 ab/(b − a). This result also matches physical limits: as the outer radius goes to infinity, the capacitance approaches that of an isolated sphere, 4π ε0 a; as the two spheres get very close (b → a), the capacitance grows without bound.

Alternatives that miss the ab product in the numerator or replace it with a single radius fail to capture how both radii interact in shaping the field and the potential, and they do not reproduce the correct limiting behavior.