Capacitance of an Isolated Spherical Conductor

Setup: Consider an isolated conducting sphere of radius a, carrying charge +q, placed in vacuum/air. Charge spreads uniformly over its outer surface.

Step 1: Potential at the surface
Treating the sphere as equivalent to a point charge q at its center (for external points):

Step 2: Apply the capacitance definition

Step 3: Simplify

Result: Capacitance is directly proportional to the radius of the sphere: C ∝ a.

If the sphere is surrounded by a medium of dielectric constant K instead of vacuum:

• C ∝ K — capacitance increases with dielectric constant.
• Capacitance is minimum in vacuum/air (K = 1) and increases in any denser dielectric medium.
• Graphs of C vs a and C vs K are both straight lines through the origin.

• Size (radius): Larger radius → larger surface area → higher capacitance.
• Medium: Higher dielectric constant → higher capacitance.
• Presence of nearby conductors: Can alter effective capacitance (not applicable for a truly isolated sphere, but relevant conceptually).

Think of capacitance like the capacity of a water tank: a wider tank (larger radius) can hold more water (charge) for the same water level (potential). Similarly, a bigger sphere stores more charge for the same potential.