AcademyCapacitors and Dielectrics
Academy
Capacitance
Level 1 - Physics topic page in Capacitors and Dielectrics.
Principle
Capacitance measures how much separated charge a conductor arrangement stores per volt.
Notation
\(C\)
capacitance
\(\mathrm{F}\)
\(Q\)
magnitude of charge on either conductor
\(\mathrm{C}\)
\(\Delta V\)
potential difference between conductors
\(\mathrm{V}\)
\(A\)
plate area
\(\mathrm{m^{2}}\)
\(d\)
plate separation
\(\mathrm{m}\)
\(\epsilon_0\)
permittivity of free space
\(\mathrm{F\,m^{-1}}\)
Method
Derivation 1: Define the storage ratio
A capacitor has two conductors carrying equal and opposite charges. The relevant voltage is the potential difference between them.
Charge-voltage ratio
\[C=\frac{Q}{\Delta V}\]
Stored charge
\[Q=C\Delta V\]
Derivation 2: Parallel-plate capacitance
Between large close plates, the field is approximately uniform away from edges. Use the field from a conducting surface and the uniform-field potential difference.
Plate charge density
\[\sigma=\frac{Q}{A}\]
Uniform field
\[E=\frac{\sigma}{\epsilon_0}=\frac{Q}{\epsilon_0A}\]
Potential difference
\[\Delta V=Ed=\frac{Qd}{\epsilon_0A}\]
Capacitance
\[C=\frac{Q}{\Delta V}=\epsilon_0\frac{A}{d}\]
Derivation 3: Isolated conducting sphere
Use the sphere's surface potential relative to infinity.
Sphere potential
\[V=\frac{1}{4\pi\epsilon_0}\frac{Q}{R}\]
Sphere capacitance
\[C=\frac{Q}{V}=4\pi\epsilon_0R\]
Rules
These are the compact capacitance relations.
Capacitance
\[C=\frac{Q}{\Delta V}\]
Charge relation
\[Q=C\Delta V\]
Parallel plates
\[C=\epsilon_0\frac{A}{d}\]
Isolated sphere
\[C=4\pi\epsilon_0R\]
Farad
\[1\,\mathrm{F}=1\,\mathrm{C\,V^{-1}}\]
Examples
Question
A capacitor stores
\[4.0\,\mu\mathrm{C}\]
at \[20\,\mathrm{V}\]
Find \(C\).Answer
\[C=\frac{Q}{\Delta V}=\frac{4.0\times10^{-6}}{20}=2.0\times10^{-7}\,\mathrm{F}\]
Checks
- Capacitance is positive and depends on geometry and material, not on the chosen charge amount.
- Use potential difference between conductors, not absolute potential.
- Larger plate area increases \(C\); larger separation decreases \(C\).
- The parallel-plate formula assumes edge effects are negligible.