Capacitance, dielectrics and stored energy

Key ideas

• A capacitor is two conducting plates separated by an insulator. Connected to a voltage, it stores equal and opposite charge on its plates.
• Capacitance is the charge stored per volt:

$Q = CV$

• $Q$ — the charge on one plate (C)
• $C$ — the capacitance (farads, F — real capacitors are µF, nF or pF)
• $V$ — the voltage across the plates (V)
• For a parallel plate capacitor:

$C = \frac{\varepsilon_0\varepsilon_r A}{d}$

• $\varepsilon_0$ — the permittivity of free space, $8.85\times10^{-12}$ F m⁻¹ (given)
• $\varepsilon_r$ — the relative permittivity of the material between the plates (no units; $1$ for air)
• $A$ — the overlap area of the plates (m²)
• $d$ — the plate separation (m)
• A dielectric (insulating material, $\varepsilon_r > 1$) between the plates increases the capacitance: its molecules polarise, partially cancelling the field, so more charge fits per volt.
• Energy stored:

$E = \tfrac{1}{2}QV = \tfrac{1}{2}CV^2$

Combining capacitors

• Parallel: capacitances simply add — more total plate area:

$C_T = C_1 + C_2 + \dots$

• Series: reciprocals add — effectively a larger gap:

$\frac{1}{C_T} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$

• This is the reverse of resistors. Parallel capacitors add; series capacitors give less than the smallest one.

Tips

• Convert µF, nF, pF before substituting ($10^{-6}$, $10^{-9}$, $10^{-12}$). Powers-of-ten slips outnumber physics errors in capacitor questions.
• For series/parallel, sanity-check the direction: parallel must give more than the largest capacitance, series must give less than the smallest.