define capacitance, as applied to both isolated spherical conductors and to parallel plate capacitors

Capacitance: The Basics ⚡️

Capacitance is a measure of how well a conductor can store electric charge. It is defined as the ratio of the charge \(Q\) on the conductor to the potential difference \(V\) between its surfaces:

\$C=\frac{Q}{V}\$

The SI unit is the farad (F). Think of a capacitor like a water tank: the charge is the water stored, and the voltage is the water pressure. The larger the tank, the more water it can hold at a given pressure – just as a larger capacitance can store more charge at a given voltage.

Isolated Spherical Conductor 🏀

For an isolated sphere of radius \(R\) in free space, the capacitance depends only on its size:

\$C{\text{sphere}} = 4\pi \varepsilon0 R\$

where \(\varepsilon_0 = 8.85\times10^{-12}\,\text{F/m}\) is the vacuum permittivity.

Example: A sphere with radius \(0.1\,\text{m}\) has

\$C = 4\pi (8.85\times10^{-12}) (0.1) \approx 1.11\times10^{-11}\,\text{F} \;(11.1\,\text{pF}).\$

Parallel Plate Capacitor 🧪

Two flat plates of area \(A\) separated by a distance \(d\) in a medium of permittivity \(\varepsilon\) form a capacitor with capacitance:

\$C_{\text{parallel}} = \varepsilon \frac{A}{d}\$

If the space between the plates is air, \(\varepsilon \approx \varepsilon_0\).

Example: Two plates \(10\,\text{cm}\times10\,\text{cm}\) (\(A=0.01\,\text{m}^2\)) separated by \(1\,\text{mm}\) (\(d=0.001\,\text{m}\)) give

\$C = (8.85\times10^{-12}) \frac{0.01}{0.001} \approx 8.85\times10^{-11}\,\text{F} \;(88.5\,\text{pF}).\$

Key Points to Remember 📌

• Capacitance is always positive because adding charge increases the potential.
• For a sphere, larger radius → larger capacitance.
• For a parallel plate capacitor, larger area or smaller separation → larger capacitance.
• Capacitance is independent of the amount of charge stored; it is a property of the geometry and material.

Quick Reference Table 📊

Practice Questions ??

1. A spherical conductor has a radius of \(0.05\,\text{m}\). Calculate its capacitance.
2. Two parallel plates of area \(20\,\text{cm}^2\) are separated by \(2\,\text{mm}\) in air. What is the capacitance?
3. Explain why a larger plate area increases capacitance using the water tank analogy.