Theory — Capacitance
1) Definition
\[
C = \frac{Q}{V}.
\]
Capacitance measures how much charge a configuration stores per volt of potential difference.
The SI unit is the farad (F):
\[
1\,\mathrm{F} = 1\,\mathrm{C/V}.
\]
2) Dielectrics
\[
\varepsilon = \varepsilon_0\,\varepsilon_r,
\qquad \varepsilon_0 \approx 8.854\times 10^{-12}\,\mathrm{F/m}.
\]
In many ideal geometries, capacitance is proportional to \(\varepsilon\), so increasing \(\varepsilon_r\) increases \(C\).
3) Common geometries
Parallel plate (uniform field approximation, fringing neglected):
\[
C = \varepsilon \frac{A}{d}.
\]
Spherical capacitor (two concentric conductors of radii \(r_1
\[
C = 4\pi \varepsilon \frac{r_1 r_2}{r_2-r_1}.
\]
Isolated conducting sphere (relative to infinity):
\[
C = 4\pi \varepsilon\,r.
\]
Coaxial cable (inner radius \(a\), outer inner radius \(b>a\), length \(L\)):
\[
C = \frac{2\pi \varepsilon L}{\ln(b/a)}.
\]
4) Using \(Q=CV\)
\[
Q = C V,
\qquad
V = \frac{Q}{C}.
\]
Once \(C\) is found from geometry, you can solve for charge at a given voltage, or voltage for a given charge.
Worked example (parallel plate)
\(A=0.1\,\mathrm{m^2}\), \(d=0.001\,\mathrm{m}\), \(\varepsilon_r=1\).
\[
C = \varepsilon_0\frac{A}{d}
= (8.854\times 10^{-12})\frac{0.1}{0.001}
\approx 8.85\times 10^{-10}\,\mathrm{F}.
\]
Notes & limitations
- Parallel-plate formula is most accurate when \(d\ll \sqrt{A}\) (small fringing).
- Real dielectrics can be non-uniform; \(\varepsilon_r\) may depend on frequency and temperature.
- Coax formula assumes a uniform dielectric between \(a\) and \(b\) and negligible end effects.
