The full, readable lecture — what a capacitor is and how it stores charge, how its capacitance depends on plate area and gap, what a dielectric does, how capacitors combine in parallel and in series, the energy locked inside a charged capacitor, the exponential charge and discharge through a resistor, and where capacitors turn up in everyday devices. As you scroll, the panel on the right plays out each idea with one everyday object you already know — water tanks filling and draining, a sponge soaking up water, a spark on discharge.
1 — What a capacitor is
A capacitor is a device that stores electric charge and electrical energy. In its simplest form it is just two parallel conducting plates separated by a small gap. Connect them to a battery and one plate collects positive charge +Q while the other collects an equal and opposite charge −Q; an electric field fills the gap between them. Think of it as a water tank — the harder you push water in (the higher the voltage), the more it holds.
Defining relationQ = C V
Q = charge on each plate (C) · V = voltage across the plates (V) · C = capacitance (F)
- Charge stored (Q) — the magnitude of charge on each plate; the two plates always carry +Q and −Q. Unit: coulomb (C).
- Capacitance (C) — how much charge a capacitor stores per volt: C = Q / V. A bigger C is a bigger "tank".
- The farad (F) — the SI unit: 1 F = 1 coulomb per volt. One farad is enormous, so real capacitors are in µF (10⁻⁶ F), nF (10⁻⁹ F) and pF (10⁻¹² F).
Key idea: capacitance is a property of the capacitor itself — its size, shape and the gap — not of how much charge you happen to put on it. Q and V both change, but their ratio C stays fixed.
2 — Capacitance & geometry
For two parallel plates in vacuum (or air), the capacitance is set entirely by their geometry: the area A of overlap of the plates and the gap d between them.
Parallel-plate capacitorC = ε₀ A / d
ε₀ = 8.85 × 10⁻¹² F/m (permittivity of free space)
A = plate area (m²) · d = separation (m)
- Bigger area A → bigger C. More plate surface holds more charge — a wider tank.
- Smaller gap d → bigger C. Closer plates pull on each other more strongly — a shallower tank fills with less push.
- C does not depend on Q or V — only on ε₀, A and d.
parallel-plate capacitance
Two plates of area 0.02 m² are 1 mm apart in air. Find C.
C = ε₀A/d = (8.85×10⁻¹² × 0.02) / (1×10⁻³)
= 1.77 × 10⁻¹⁰ F ≈ 177 pF
Real life: to pack a big capacitor into a tiny case, makers use a huge area (long foil strips rolled up) with a very thin gap — exactly what C = ε₀A/d tells you to do.
3 — Dielectrics
Slide an insulating slab — a dielectric (mica, glass, plastic, ceramic, paper) — into the gap and the capacitance increases by a factor called the relative permittivity (or dielectric constant), εᵣ. The slab's molecules line up in the field and partly cancel it, so the same charge sits at a lower voltage — meaning C = Q/V is larger.
With a dielectricC = εᵣ ε₀ A / d = εᵣ × C₀
εᵣ = relative permittivity (no units; εᵣ > 1 always)
| Dielectric | εᵣ (approx) |
|---|
| Vacuum / air | 1.0 |
| Paper | 3.5 |
| Mica | 6 |
| Water | 80 |
Sponge picture: drop a sponge in a tank and it holds far more water for the same depth. A dielectric does the same for charge — for the same voltage, the capacitor holds εᵣ times more. The slab also lets the plates sit closer without sparking, raising C further.
4 — Capacitors in parallel
Capacitors in parallel are connected across the same two points, so each feels the same voltage V. Their plate areas effectively add together, so the total capacitance is just the sum.
Parallel combinationC = C₁ + C₂ + C₃ + …
Same V across each · total charge Q = Q₁ + Q₂ + …
- Voltage is the same across every capacitor.
- Charges add: the bank stores more than any single one.
- The combined capacitance is larger than the biggest single capacitor.
parallel combination
Find the total capacitance of 2 µF and 3 µF in parallel.
C = C₁ + C₂ = 2 + 3 = 5 µF
Tank picture: stand two tanks side by side and link their tops — the total water store is simply the sum of the two. Parallel = more total capacity.
5 — Capacitors in series
Capacitors in series are connected one after another in a chain. The same charge Q sits on every capacitor, but the supply voltage splits between them. The combination behaves like a single capacitor with a larger effective gap, so its capacitance is smaller than any single member.
Series combination1/C = 1/C₁ + 1/C₂ + 1/C₃ + …
Same Q on each · total V = V₁ + V₂ + …
(two only) C = C₁C₂ / (C₁ + C₂)
- Charge Q is the same on every capacitor in the chain.
- Voltages add up to the supply voltage.
- The combined capacitance is smaller than the smallest single capacitor.
series combination
Find the total capacitance of 2 µF and 3 µF in series.
C = C₁C₂/(C₁+C₂) = (2×3)/(2+3) = 6/5 = 1.2 µF
Tank picture: chain the tanks nose-to-tail and the same trickle passes through each — the combined store is smaller than either tank alone. Series = less total capacity, but it shares the voltage.
6 — Energy stored
Charging a capacitor takes work, because each extra bit of charge must be pushed onto a plate that already repels it. That work is stored as electrical potential energy in the field between the plates — the "stored water" of the tank. Discharge it suddenly and the energy comes out all at once, often as a bright spark.
Energy in a charged capacitorE = ½ C V² = ½ Q V = Q² / (2C)
E in joules (J) · all three forms are equal (using Q = CV)
| Quantity | While charging |
|---|
| Charge Q | builds up linearly with V |
| Energy E | builds up with the square of V (E ∝ V²) |
energy stored
A 100 µF capacitor is charged to 50 V. How much energy is stored?
E = ½CV² = ½ × 100×10⁻⁶ × 50²
= ½ × 100×10⁻⁶ × 2500 = 0.125 J
Why ½? The voltage rises from 0 to V as it charges, so the average voltage during charging is V/2 — hence E = ½QV, not QV. Doubling the voltage quadruples the stored energy.
7 — Charging & discharging through a resistor
Put a resistor R in series with the capacitor and the charging is no longer instant. The charge climbs quickly at first, then ever more slowly, approaching the full charge along a smooth exponential curve — exactly like water filling a tank: a fast initial rush, then a gentle topping-up. The product RC = τ (tau) is the time constant, the natural timescale of the circuit.
Charging & dischargingcharging: Q = Q₀ (1 − e^(−t/RC))
discharging: Q = Q₀ e^(−t/RC)
time constant τ = R C (seconds)
- After one time constant (t = RC) the capacitor reaches ≈ 63% of full charge.
- After about 5 time constants it is essentially fully charged (≈ 99%).
- Discharging mirrors it: the charge falls fast then slows, dropping to ≈ 37% after one τ.
- Bigger R or bigger C → a longer τ → slower charge and discharge.
Real life: this RC timing runs blinking indicators, camera-flash recharge "ready" delays and the fade of a touch lamp. The same curve fills your tank and drains it.
8 — Applications & recap
Capacitors are everywhere because they can store energy and release it quickly, smooth out changing voltages, and respond to touch:
- Camera flash — a capacitor charges slowly from a small battery, then dumps its energy into the flash tube in a burst of light.
- Power-supply smoothing — a large capacitor fills the dips in a rectified supply, turning bumpy voltage into a steady DC.
- Touchscreens — your finger changes the local capacitance of a tiny grid; the phone senses exactly where you touched.
- Defibrillator — a capacitor stores a large charge and delivers a controlled, life-saving pulse to the heart.
quick recap numerical
A 200 µF capacitor charges to 12 V. Find Q and the stored energy E.
Q = CV = 200×10⁻⁶ × 12 = 2.4 × 10⁻³ C
E = ½CV² = ½ × 200×10⁻⁶ × 144 = 0.0144 J
- Capacitor stores charge: Q = CV; capacitance C = Q/V, unit farad.
- Parallel-plate: C = ε₀A/d — more area or smaller gap → bigger C.
- A dielectric multiplies C by εᵣ: C = εᵣε₀A/d.
- Parallel: C = C₁ + C₂ + …; series: 1/C = 1/C₁ + 1/C₂ + …
- Stored energy: E = ½CV² = ½QV (E ∝ V²).
- Through R: charge/discharge is exponential with time constant τ = RC.