Capacitance — Physics II Reference

00

What Are We Building?

A capacitor is a charge storage device. But that's the boring version. The deeper truth: a capacitor stores energy in the electric field between two conductors. The geometry of those conductors — area, separation, shape — determines how much energy fits.

The core chain

Geometry → Electric Field → Potential Difference → Capacitance → Stored Energy

Everything on this page flows from that chain. Once you see it, the formulas stop being things to memorize and start being things you could re-derive in a pinch.

01

Definition of Capacitance

We define capacitance as the ratio of charge stored to voltage applied:

Fundamental Definition

\[ C \;=\; \frac{Q}{V} \]

CCapacitance — "how good it is at storing charge" [Farads, F] QCharge separated onto the plates [Coulombs, C] VVoltage (potential difference) across the plates [Volts, V]

Feynman reframe

Rearrange to \(Q = CV\). For a fixed voltage, bigger \(C\) means you can jam more charge onto the plates. Capacitance literally measures: "how much charge per volt?"

Notice that \(C\) depends only on geometry — not on \(Q\) or \(V\) individually. Double the voltage and you double the charge, but their ratio stays fixed. That fixed ratio is a pure property of the shape and size of the conductors.

02

The Parallel-Plate Capacitor

The simplest geometry — two flat conducting plates of area \(A\), separated by distance \(d\). Let's derive its capacitance from first principles.

Parallel plate capacitor — field lines point from +Q to −Q plate

Step-by-step derivation

① E

From Gauss's Law, the electric field between infinite parallel plates carrying surface charge density \(\sigma = Q/A\) is uniform:

\[ E \;=\; \frac{\sigma}{\varepsilon_0} \;=\; \frac{Q}{\varepsilon_0 A} \]

The field is uniform (same everywhere between plates) — that's what makes parallel plates so clean mathematically. Every field line is parallel and evenly spaced.

② V

Voltage is work done per unit charge to move across the gap. For a uniform field, it's simply \(E\) times the distance:

\[ V \;=\; E \cdot d \;=\; \frac{Q \, d}{\varepsilon_0 A} \]

Intuition: bigger gap \(d\) → more work to cross → higher voltage for same charge.

③ C

Now plug into the definition \(C = Q/V\):

\[ C \;=\; \frac{Q}{V} \;=\; \frac{Q}{\dfrac{Qd}{\varepsilon_0 A}} \;=\; \varepsilon_0 \frac{A}{d} \]

\(Q\) cancels — confirming \(C\) is purely geometric.

Parallel Plate Result

\[ \boxed{C \;=\; \varepsilon_0 \frac{A}{d}} \]

ε₀Permittivity of free space — \(8.854 \times 10^{-12}\) F/m APlate area [m²] — more area → more field lines → larger C dPlate separation [m] — larger gap → harder to maintain field → smaller C

Geometric intuition

Think of the field between plates like a stretched spring. Bigger plates give more "springs" in parallel → more force → more storage. Bigger gap stretches each spring further → same charge creates less field density → less storage per volt.

03

Energy Stored in a Capacitor

To charge a capacitor you have to do work — pushing positive charge onto an already positive plate fights the repulsion. That work gets stored as energy in the electric field.

Deriving the ½

At any instant during charging, suppose charge \(q\) is already on the plates. The voltage across the capacitor at that moment is \(v = q/C\). The tiny bit of work to add another infinitesimal charge \(dq\) is:

\[ dU \;=\; v \, dq \;=\; \frac{q}{C} \, dq \]

Integrate from \(q = 0\) to \(q = Q\):

\[ U \;=\; \int_0^Q \frac{q}{C} \, dq \;=\; \frac{1}{C} \cdot \frac{q^2}{2} \Bigg|_0^Q \;=\; \frac{Q^2}{2C} \]

Substituting \(Q = CV\) gives the three equivalent forms:

Energy Stored — Three Forms

\[ U \;=\; \frac{Q^2}{2C} \;=\; \frac{1}{2}\,C V^2 \;=\; \frac{1}{2}\,Q V \]

Why ½?

The voltage starts at 0 and ramps up to \(V\) as charge accumulates. The average voltage doing work is \(V/2\). So \(U = Q \cdot V_{\text{avg}} = Q \cdot V/2 = \frac{1}{2}QV\). The ½ is the signature of any quadratic storage — you see the same factor in \(\frac{1}{2}mv^2\) and \(\frac{1}{2}kx^2\).

Energy lives in the field

For a parallel plate capacitor, the energy density (energy per unit volume) in the field is:

Electric Field Energy Density

\[ u \;=\; \frac{U}{A\,d} \;=\; \frac{1}{2}\,\varepsilon_0 E^2 \]

This is profound: the energy isn't sitting on the plates — it's distributed throughout the electric field. Stronger field → more energy per unit volume. This viewpoint (field carries energy) generalizes to all of E&M, including electromagnetic waves.

04

Series & Parallel Combinations

Parallel (left) vs. Series (right) — note which quantity is shared

Parallel — same voltage, charges add

In parallel, both plates of every capacitor share the same two nodes, so they all see the same voltage \(V\). Each capacitor stores its own charge \(Q_i = C_i V\). The total charge drawn from the source:

\[ Q_{\text{total}} = Q_1 + Q_2 + \cdots \;\Rightarrow\; C_{\text{eq}} V = C_1 V + C_2 V + \cdots \]

Dividing through by \(V\):

Parallel Combination

\[ \boxed{C_{\text{eq}} \;=\; C_1 + C_2 + C_3 + \cdots} \]

Geometric picture

Putting capacitors in parallel is like stacking their plates side by side — you're effectively adding plate area. Recall \(C \propto A\). More area, bigger capacitance. Direct sum.

Series — same charge, voltages add

In series, only one wire connects adjacent capacitors. Whatever charge flows onto the first plate must come from the second, and so on — every capacitor holds the same charge \(Q\). The voltages, however, split across each one: \(V_i = Q/C_i\). Total voltage:

\[ V_{\text{total}} = V_1 + V_2 + \cdots \;\Rightarrow\; \frac{Q}{C_{\text{eq}}} = \frac{Q}{C_1} + \frac{Q}{C_2} + \cdots \]

Dividing through by \(Q\):

Series Combination

\[ \boxed{\frac{1}{C_{\text{eq}}} \;=\; \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \cdots} \]

Geometric picture

Series is like stacking the gaps — you're effectively increasing the plate separation. Recall \(C \propto 1/d\). More total gap → smaller capacitance. The reciprocal rule is the algebraic mirror of adding distances.

Configuration	Shared Quantity	Summed Quantity	Effect on C
Parallel	Voltage \(V\)	Charge \(Q\)	↑ Increases
Series	Charge \(Q\)	Voltage \(V\)	↓ Decreases

05

Unit Anatomy

Start from the definition \(C = Q/V\) and unpack every unit layer:

[C] = C / V — charge over voltage

V = J/C — volts are joules per coulomb

→ C / (J/C) = C² / J — expanding

J = kg·m²/s² — SI energy

→ C²·s² / (kg·m²) = 1 Farad (F)

Why farads are huge

1 F = 1 C²·s²/(kg·m²). A coulomb is already an enormous amount of charge (~\(6.24 \times 10^{18}\) electrons). Real capacitors are measured in microfarads (μF, \(10^{-6}\)) or picofarads (pF, \(10^{-12}\)). Supercapacitors can reach farads — that's why they're used as batteries in some applications.

We can also verify \(\varepsilon_0 A/d\) has units of farads:

Dimensional check

\[ [\varepsilon_0] = \frac{\text{C}^2}{\text{N·m}^2} = \frac{\text{C}^2 \cdot \text{s}^2}{\text{kg} \cdot \text{m}^3} \;\Rightarrow\; \varepsilon_0 \frac{A}{d} = \frac{\text{C}^2 \cdot \text{s}^2}{\text{kg} \cdot \text{m}^3} \cdot \frac{\text{m}^2}{\text{m}} = \frac{\text{C}^2 \cdot \text{s}^2}{\text{kg} \cdot \text{m}^2} = \text{F} \;\checkmark \]

06

Where This Fits

Zoom out. You're assembling the full energy-storage toolkit of E&M:

Element	Stores	Key Equation	Field
Capacitor	Energy in \(\mathbf{E}\)-field	\(U = \frac{1}{2}CV^2\)	Electric
Inductor	Energy in \(\mathbf{B}\)-field	\(U = \frac{1}{2}LI^2\)	Magnetic
Resistor	Dissipates energy	\(P = I^2 R\)	None (heat)

The RLC analogy

In an RLC circuit, the capacitor and inductor swap energy back and forth (electric ↔ magnetic), while the resistor bleeds it away as heat — exactly like a mass-spring-damper swapping potential and kinetic energy. The math is identical. That's not a coincidence — it's the same differential equation.

The deeper pattern: wherever you see \(\frac{1}{2}(\text{property})(\text{variable})^2\), you're looking at quadratic energy storage. Springs, capacitors, inductors, acoustic resonators — all the same skeleton.