Capacitance and Dielectrics
A capacitor is nothing more than two conductors holding equal and opposite charge, separated by some gap — capacitance C=Q/V is just the fixed ratio between how much charge it holds and the potential difference that results, a purely geometric property like the field-integral results from Units 201-202. What makes capacitors useful is that they store energy in the field itself, and inserting an insulating dielectric between the plates lets them store substantially more of it in the same physical space.
What you'll learn
- Define capacitance and calculate it for a parallel-plate capacitor from its geometry.
- Calculate the energy stored in a charged capacitor.
- Analyze capacitors combined in series and in parallel.
- Explain how inserting a dielectric changes a capacitor's capacitance, and calculate the new value.
- Determine how charge and voltage change when a dielectric is inserted, with the battery connected vs. disconnected.
- Calculate the energy density stored in an electric field.
1. Capacitance and the Parallel-Plate Capacitor
Any two conductors holding equal and opposite charge, separated by an insulating gap, form a capacitor. The ratio of charge to potential difference, C=Q/V, is fixed by the geometry alone — it doesn't change as the capacitor charges or discharges, only Q and V do (in lockstep, keeping their ratio constant).
For a parallel-plate capacitor with plate area A and separation d, C=ε₀A/d — bigger plates or a smaller gap both increase capacitance, since either one lets more charge accumulate for the same field strength (and hence the same voltage, since V=Ed for a uniform field).
2. Series and Parallel Combinations
Capacitors combine with rules that are the mirror image of resistors: in series, 1/C_eq=Σ1/Cᵢ (equivalent capacitance is *smaller* than the smallest individual value); in parallel, C_eq=ΣCᵢ (equivalent capacitance is larger than any individual value). It's worth deliberately pausing on this every time, since the resistor instinct runs the opposite direction.
Physically, series capacitors share the same charge Q (each plate pair induces an equal charge on its neighbor) but split the total voltage; parallel capacitors share the same voltage but hold different charge according to their individual C.
3. Energy Storage and Dielectrics
A charged capacitor stores energy U=½QV=½CV²=Q²/(2C). The factor of ½ reflects that charge builds up gradually — the first bit of charge faces almost no opposing voltage, while the last bit is pushed against the nearly-final voltage, and integrating that process (not simply multiplying final Q by final V) is what produces the ½.
Inserting a dielectric — an insulating material — between the plates multiplies capacitance by its dielectric constant κ (always >1), because the dielectric's own polarization partially cancels the field between the plates, letting more charge accumulate for the same voltage. What happens to Q and V individually when a dielectric is inserted depends on whether a battery is still connected: fixed V (battery connected) means Q grows; fixed Q (isolated) means V shrinks.
Key equations
- C = Q/V — The defining relationship — a fixed property of the capacitor's geometry, not of how much charge happens to be on it at a given moment.
- C = ε₀A/d — Capacitance of a parallel-plate capacitor with vacuum (or air) between the plates.
- Series: 1/C_eq=Σ1/Cᵢ. Parallel: C_eq=ΣCᵢ — Combining capacitors — note this is the *opposite* pattern from resistors: capacitors in series combine like resistors in parallel, and vice versa.
- U = ½QV = ½CV² = Q²/(2C) — Energy stored in a charged capacitor — the factor of ½ comes from the charge building up gradually (the first charge placed on the plate does no work against opposing charge, the last increment does the most).
- C = κC₀, u = ½ε₀E² — A dielectric filling the gap multiplies capacitance by κ. Energy density u gives the energy stored per unit volume anywhere a field E exists — capacitor energy is just this integrated over the field's volume.