Integrate a spherical capacitor from two energy accounts

Integrate the gap field

For concentric conductors of radii a and b carrying ±Q, the vacuum gap field is kQ/r². The inner-minus-outer potential difference is integral from a to b of E dr = kQ(1/a − 1/b). The material fields are zero. This derivation fixes the sign and reference before forming a capacitance.

Form the geometry coefficient

C = Q/delta V = 4pi epsilon0 ab/(b − a). Charge cancels for this linear geometry. Increasing the outer radius at fixed a reduces capacitance toward 4pi epsilon0 a; the limit describes an isolated sphere relative to infinity. A thin gap approaches the local parallel-plate form with area near 4pi a², but not an arbitrary flat-plate substitution at wide gaps.

Integrate the field energy

The optional local-energy check uses u = epsilon0 E²/2 and spherical volume element 4pi r² dr. Its integral equals 2pi epsilon0 k²Q²(1/a − 1/b) = Q²/(2C). This agrees with reversible charging work integral q/C dq. These are independent derivations within one model, not independent physical measurements.

Worked original case

For a = 1 m, b = 2 m and Q = 2 nC, C ≈ 222.530 pF, delta V ≈ 8.98755 V and U ≈ 8.98755 nJ. Doubling both radii at fixed ratio doubles C; at fixed Q it halves voltage and stored energy.

Investigation

Derive both integrals before checking three radius pairs. Keep the fixed-Q constraint explicit while comparing energy. Test the outer-radius limit and the thin-gap approximation, stating what dimensionless ratio controls each.

Changed-case practice

Sources and boundaries

Original cases checked against the official framework on 30 October 2026, with OpenStax verification on 31 October 2026 against the current official framework, Unit 10 and OpenStax capacitance. These are virtual supplied models, not instructions to charge, ground, wire or handle real equipment. Public answers are practice; retained plans and self-reports do not establish independent mastery, actual laboratory participation or AP authorization.