Capacitance of an isolated conducting sphere in vacuum What is the capacitance C of a single isolated conducting sphere of radius R in vacuum (permittivity ε0)?

Introduction / Context:
An isolated conducting sphere is a fundamental electrostatics configuration. Its capacitance defines how much charge it can store per unit potential relative to infinity. This result is widely used in high-voltage engineering, lightning protection, and as a building block for more complex capacitance calculations.

Given Data / Assumptions:

• Perfect conductor shaped as a sphere of radius R.
• Surrounded by vacuum (permittivity ε0), reference potential at infinity.
• Electrostatic steady state; no nearby conductors or dielectrics.

Concept / Approach:

For a charged isolated sphere, the electric field outside is identical to that of a point charge at the center. The potential at the surface relative to infinity is V = (1 / (4π ε0)) * (Q / R). The capacitance is C = Q / V, yielding C = 4π ε0 R, linear in the radius.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional check: ε0 has units C^2/(N·m^2); multiplying by length R gives farads, as required. The result also matches limiting cases for large R (larger C) and small R (smaller C).

Why Other Options Are Wrong:

2π ε0 R is off by a factor 2; 4π ε0 R^2 has wrong length dependence; 4π lacks units and parameters.

Common Pitfalls:

Confusing the sphere–plane or sphere–shell capacitance with the isolated sphere value; nearby conductors change the effective capacitance.

Final Answer:

4π ε0 R