Understanding the capacitance of an isolated sphere is essential for analysing how charged conductors store electrical energy and develop potential, forming a cornerstone of electric field and potential theory.

Capacitance of an Isolated Sphere

In this subsubtopic, we examine how an isolated conducting sphere behaves as a simple capacitor, storing charge and developing a measurable electric potential relative to infinity. The OCR specification explicitly requires students to derive the expression C = 4πɛ₀R, using the known relationship between charge and potential for a spherical conductor. This derivation links electric fields, electric potential, and capacitance into a single, coherent result important for understanding more complex capacitive systems.

Properties of a Conducting Sphere

A conducting sphere distributes any excess charge uniformly on its outer surface. This uniform distribution arises because mobile charges within the conductor repel one another and settle in a configuration that minimises their potential energy. The symmetry of the sphere allows us to model the entire charge as if it were concentrated at the centre when analysing the electric potential outside the sphere. Outside a uniformly charged conducting sphere, the electric field is radial and identical to that of a point charge located at the centre.

Radial electric field lines of a single positive charge. Outside a uniformly charged conducting sphere the field has the same radial form, decreasing in magnitude with distance. This supports the use of a point-charge potential when deriving the capacitance expression. Source.

Because the reference potential is defined as zero at infinity, the potential of the sphere depends solely on the amount of charge it carries and its radius.

After establishing how charge distributes, we can consider how electric potential behaves around the sphere and how this leads to its capacitance.

Electric Potential of a Charged Sphere

For points outside a uniformly charged spherical conductor, the potential is identical to that produced by a point charge. This allows us to use the familiar expression for the electric potential due to a point charge Q at a distance r from the centre of the sphere. At the sphere’s surface, r = R, and therefore we can express the potential directly in terms of the sphere’s radius and total charge.

EQUATION
—-----------------------------------------------------------------
Electric Potential of a Sphere (V) = Q / (4πɛ₀R)
Q = Charge on the sphere (C)
ɛ₀ = Permittivity of free space (F m⁻¹)
R = Radius of sphere (m)
—-----------------------------------------------------------------

This relationship is essential because capacitance links charge and potential, and for an isolated sphere both these quantities are straightforward to define.

Between the above result and the definition of capacitance, we now have the tools needed to derive the required expression.

Deriving the Capacitance Expression

The definition of capacitance for any conductor is given by C = Q/V. For a spherical conductor, substituting the potential expression into this definition allows us to obtain a general expression for its capacitance. The derivation relies on straightforward substitution and algebra and makes use of the fact that the potential of the sphere is entirely determined by its surface.

EQUATION
—-----------------------------------------------------------------
Capacitance of a Sphere (C) = 4πɛ₀R
C = Capacitance (F)
ɛ₀ = Permittivity of free space (F m⁻¹)
R = Radius of sphere (m)
—-----------------------------------------------------------------

This result demonstrates that capacitance depends only on the physical size of the conductor and the electric properties of the surrounding medium. It also highlights that a larger radius corresponds to a larger capacitance, a concept that underpins the design of many practical capacitors and the behaviour of natural charged objects such as planets and clouds.

The derivation is therefore a powerful example of how electrostatic principles combine to give meaningful physical results.

Taking the limit of the spherical-capacitor expression as R2→∞R_2 \to \inftyR2​→∞ gives the isolated sphere result C=4πε0RC = 4\pi\varepsilon_0 RC=4πε0​R.

Cross-section of a spherical capacitor with inner radius R₁ and outer radius R₂, showing radial electric field lines and a Gaussian surface. Letting R₂ approach infinity reduces the arrangement to a single isolated sphere, yielding the required capacitance expression. Extra detail is present (outer sphere and Gaussian surface) to illustrate the limiting argument. Source.

Key Features of the Isolated Sphere Model

The simple expression for capacitance reveals several important characteristics of an isolated conducting sphere:

• Capacitance depends solely on geometry, not on the amount of charge stored. This distinguishes the concept of capacitance from charge itself.

• A larger sphere has a greater capacitance, meaning it can hold more charge for the same potential difference.

• The surrounding space acts as the dielectric, since the isolated sphere is assumed to be in free space with permittivity ɛ₀.

• The potential is referenced to infinity, ensuring consistency with the definition of electric potential for isolated systems.

• Charge distribution remains uniform, ensuring the field outside the sphere corresponds to that of a point charge.

These features make the isolated sphere a fundamental model for understanding more complex capacitance systems encountered later in physics.

Applications of the Isolated Sphere Concept

Although an isolated spherical conductor is an idealised system, the underlying principles support a variety of real-world contexts:

• Large conducting bodies, such as planets, often behave approximately as isolated charged spheres for electrostatic modelling.

• Spherical capacitors, though more advanced, build on the same principles of symmetry and radial electric fields.

• High-voltage equipment, including Van de Graaff generators, relies on the relationship between radius and capacitance to accumulate large charge without excessive potential rise.

• Understanding the limits of charge storage on conductors becomes easier when capacitance is directly linked to physical size.

These conceptual applications reinforce why deriving C = 4πɛ₀R is central to the curriculum and provides essential grounding for topics involving electric potential and energy.