Spherical and Cylindrical Capacitors

Deriving capacitance for curved conductor geometries using Gauss's Law.

Spherical Capacitor

A spherical capacitor consists of two concentric conducting spherical shells — an inner shell of radius \( a \) carrying charge \( +Q \) and an outer shell of radius \( b \) carrying charge \( -Q \).

Electric Field in the Gap

By Gauss's Law, the field in the region \( a \lt r \lt b \) is the same as if all the charge were at the centre:

\[ \vec{E} = \frac{Q}{4\pi\varepsilon_0 r^2} \vec{r} \]

The field is zero for \( r \lt a \) (inside the inner conductor) and for \( r \gt b \) (the outside field due to \( +Q \) and \( -Q \) cancels).

Differential displacement \( d\vec{r} \)

To use \( dV = - \vec{E} \cdot d\vec{r} \), we take differential displacement \( d\vec{r} \) as a general (variable) displacement along \( d\vec{r} \).

\[ d\vec{r} = dr \vec{r} \]

The actual direction will be determined by limits of integration and we don't have to change anything with direction of \( d\vec{r} \) as we are using algebraic dot product of vectors.

Potential Difference

\[ V(b) - V(a) = -\int_a^b E\, dr = -\frac{Q}{4\pi\varepsilon_0}\int_a^b \frac{dr}{r^2} = \frac{Q}{4\pi\varepsilon_0}\left(\frac{1}{b} - \frac{1}{a}\right) \]

\( V(b) - V(a) \) comes out to be negative. This is because \( V(b) \lt V(a) \) as B sphere has \( -Q \) charge and \( b \gt a \). But, for capacitance, we only take positive potential difference \( |V(b) - V(a)| \).

\[ V_{\text{Capacitance}} = \frac{Q}{4\pi\varepsilon_0}\left(\frac{1}{a} - \frac{1}{b}\right) \]

Capacitance

\[ \boxed{C = \frac{4\pi\varepsilon_0 ab}{b - a}} \]

Special cases:

If \( b \to \infty \): \( C \to 4\pi\varepsilon_0 a \) — the self-capacitance of an isolated sphere.
As \( b - a = d \to 0 \) with \( a \approx b \approx R \): \( C \approx 4\pi\varepsilon_0 R^2/d = \varepsilon_0 A/d \) — reduces to the parallel plate formula (locally flat geometry).

The \( b \to \infty \) result is physically important: it means even an isolated sphere has a capacitance — the second "plate" is conceptually at infinity with potential zero.

Cylindrical Capacitor

A cylindrical capacitor consists of two coaxial conducting cylinders — inner cylinder of radius \( a \), outer cylinder of radius \( b \), both of length \( L \) (where \( L \gg b \) to treat end effects as negligible).

Field in the Gap

Using a Gaussian cylinder of radius \( r \) (\( a \lt r \lt b \)) and length \( l \):

\[ E \cdot 2\pi r l = \frac{\lambda l}{\varepsilon_0} \implies E = \frac{\lambda}{2\pi\varepsilon_0 r} \]

where \( \lambda = Q/L \) is the linear charge density.

Potential Difference

\[ V = \int_a^b E\, dr = \frac{\lambda}{2\pi\varepsilon_0} \ln\left(\frac{b}{a}\right) \]

Capacitance

\[ \boxed{C = \frac{2\pi\varepsilon_0 L}{\ln(b/a)}} \]

The capacitance per unit length is \( C/L = 2\pi\varepsilon_0 / \ln(b/a) \), a useful quantity for coaxial cables.

Comparison of Geometries

Geometry	Field Law Used	Capacitance
Parallel Plate	Gauss (planar)	\( \varepsilon_0 A/d \)
Spherical	Gauss (spherical)	\( 4\pi\varepsilon_0 ab/(b-a) \)
Cylindrical	Gauss (cylindrical)	\( 2\pi\varepsilon_0 L/\ln(b/a) \)

Geometric Pattern Behind All Capacitance Formulae

Across all geometries studied so far, the derivation follows the same structural steps:

Use Gauss's Law to determine the electric field \( E(r) \) from symmetry.
Compute the potential difference using \( \Delta V = -\int \vec{E} \cdot d\vec{r} \).
Use \( C = \dfrac{Q}{\Delta V} \).

The crucial observation is this: the geometry enters only through the integral that produces the potential difference.

How Geometry Appears in Each Case

Parallel Plate:

\[ E = \frac{\sigma}{\varepsilon_0} \quad \Rightarrow \quad \Delta V = Ed \propto d \] \[ C = \frac{\varepsilon_0 A}{d} \]

Capacitance is inversely proportional to the linear separation \( d \).

Cylindrical:

\[ E = \frac{\lambda}{2\pi\varepsilon_0 r} \quad \Rightarrow \quad \Delta V \propto \ln\!\left(\frac{b}{a}\right) \] \[ C = \frac{2\pi\varepsilon_0 L}{\ln(b/a)} \]

Capacitance is inversely proportional to the logarithmic radial factor.

Spherical:

\[ E = \frac{Q}{4\pi\varepsilon_0 r^2} \quad \Rightarrow \quad \Delta V \propto \left(\frac{1}{a} - \frac{1}{b}\right) \] \[ C = \frac{4\pi\varepsilon_0 ab}{b-a} \]

Equivalently,

\[ \frac{1}{C} = \frac{1}{4\pi\varepsilon_0} \left(\frac{1}{a} - \frac{1}{b}\right) \]

Here, capacitance is inversely proportional to the difference of inverse radii.

In every geometry, capacitance is the inverse of the geometric factor that appears in the potential integral. The electric field determines how potential accumulates with distance, and that accumulation pattern determines \( C \).

Thus, capacitance is fundamentally a measure of how geometry controls the rate at which potential rises with charge.