A cylindrical capacitor has an outer radius R and a...

A cylindrical capacitor has an outer radius $R$ and a separation $d$ between the cylinders. Determine what the capacitance approaches in the limit where $d \ll R$. (Hint: Express the capacitance in terms of the ratio $d / R$ and then examine what happens as that ratio becomes very small compared to 1.) Explain why the limit on the capacitance makes sense.

Key Concepts

Small Separation Approximation

When the gap d between the cylinders is much smaller than the radius R (d ? R), the logarithmic term in the capacitance expression can be approximated using a series expansion (for example, using ln(1 + x) ? x for small x). This mathematical simplification allows the capacitor's behavior to be characterized in a limit where the curved geometry closely approximates a simpler configuration.

Parallel Plate Capacitor Analogy

In the limit where the separation is much less than the radius, the capacitor behaves similarly to a parallel plate capacitor because locally the curved surfaces appear nearly flat. As a result, the capacitance per unit length simplifies to an expression that is inversely proportional to the separation, mirroring the dependence seen in parallel plate capacitors. This analogy helps validate the limiting behavior observed in the full cylindrical capacitor model.

Cylindrical Capacitor Geometry

A cylindrical capacitor consists of two coaxial cylinders with a dielectric medium between them. The capacitance depends on the radii of the two cylinders and the distance separating them. This geometry is a fundamental configuration in electrostatics that illustrates how curved surfaces interact with electric fields.

Capacitance Expression for Coaxial Cylinders

In a cylindrical capacitor, the capacitance per unit length is typically given by an expression involving the logarithm of the ratio of the outer and inner radii. This logarithmic dependence directly ties the capacitor’s storage capability to its geometric parameters, reflecting how the electric field is distributed between the curved surfaces.

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