Capacitors & Capacitance

A capacitor is two conducting plates separated by an insulator. Connect a battery and one plate piles up +Q, the other an equal −Q; the device stores charge and electrical energy.

Picture a water tank: the battery's voltage is the water pressure, the charge is the water stored. A bigger pressure (V) pushes in more water (Q), and a wider tank holds more water per unit pressure — that "wideness" is the capacitance C.

For two parallel plates of area A a distance d apart in vacuum, capacitance depends only on geometry:

• Bigger area A — more room for charge to spread → C increases.
• Smaller gap d — plates pull harder on each other's charge → C increases.

Slide an insulator — a dielectric — between the plates and the capacitance jumps. The dielectric's molecules polarise, partly cancelling the field, so the same voltage now stores more charge.

Think of a dry sponge dropped into the tank: it soaks up field and lets you pack far more charge into the same space — that is why every practical capacitor has a dielectric filling.

Connect capacitors side by side across the same two wires and every one feels the same voltage V. The total charge is the sum, so the capacitances simply add.

Like placing two water tanks next to each other under one pressure pipe: their storage volumes add directly — parallel always gives more total capacitance.

Connect capacitors in a chain and the same charge Q sits on every one, while the battery's voltage splits between them. The reciprocals add, so the combined capacitance is smaller than the smallest.

Charging a capacitor takes work, pushing charge against the growing voltage. That work is stored in the electric field and can be released in an instant — a bright spark.

Because the energy goes as V squared, doubling the voltage stores four times the energy — which is exactly why a camera flash charges to a high voltage before firing.

Put a resistor in the path and the capacitor cannot fill instantly. The voltage rises fast at first, then ever more slowly as it approaches the battery voltage — an exponential curve.

Like filling a tank through a narrow tap: a near-empty tank fills quickly, but as it nears the top the back-pressure rises and the inflow tapers off.

Disconnect the battery and let the capacitor empty through the resistor: the voltage falls quickly at first, then trails off, a mirror image of charging.

This steady, predictable drain is the heartbeat of timers and the smooth fade of camera-flash recharge lights — change R or C and you tune exactly how long it takes.

You have walked through the whole story of capacitors — from a single pair of plates to charging and discharging through a resistor.

• Camera flash — a capacitor charges slowly, then dumps E = ½CV² into the bulb as a brilliant burst.
• Power-supply smoothing — large capacitors fill in the dips between AC peaks, giving steady DC.
• Defibrillator — stores a big charge, then releases it through the heart in a controlled jolt.

1. Q = C V; capacitance is measured in farads (C/V) and is fixed by geometry.
2. Parallel-plate: C = ε₀A/d — bigger area or smaller gap raise C.
3. A dielectric multiplies C by εᵣ: C = εᵣε₀A/d.
4. Parallel: C = C₁ + C₂ (bigger). Series: 1/C = 1/C₁ + 1/C₂ (smaller).
5. Stored energy E = ½CV² = ½QV = Q²/2C.
6. Charging V = V₀(1 − e^(−t/RC)); discharging V = V₀e^(−t/RC); τ = RC.