Electric Potential and Capacitors

Lifting a book gives it gravitational potential energy. In the same way, pushing a positive charge toward another positive charge takes work and gives it electric potential energy. Electric potential is that energy per coulomb, and the voltage between two points is the work needed per coulomb to move charge from one to the other:

V = W ÷ q

So 1 volt means 1 joule per coulomb. A 9 V battery gives each coulomb that passes through it 9 J of energy.

Around a point charge, the potential is the same at every point the same distance away, so the equipotentials are circles (spheres in 3D). Moving a charge along an equipotential takes no work. Field lines always cross equipotentials at right angles, pointing from high potential to low.

Connect two flat, parallel metal plates to a battery and one becomes positive, the other negative. Between them, away from the edges, the electric field is uniform: the same strength and direction everywhere. Its strength is

E = V ÷ d

where d is the gap. That's why field strength can be written in volts per meter, which is the same unit as newtons per coulomb. The equipotentials between the plates are flat and evenly spaced. Halve the gap and the field doubles.

A pair of plates like this is a capacitor. When it's connected to a battery, electrons are pulled off one plate and pushed onto the other, until the voltage across the plates matches the battery. One plate ends up with charge +Q and the other −Q. The charge stored is proportional to the voltage, and the ratio is the capacitance:

C = Q ÷ V

For a parallel-plate capacitor with plate area A and gap d,

C = ε₀A ÷ d, with ε₀ = 8.85 × 10⁻¹² F/m

Bigger plates and a smaller gap store more charge per volt. Filling the gap with an insulating dielectric, such as plastic or ceramic, multiplies the capacitance by the material's dielectric constant κ. The molecules in the dielectric line up slightly in the field, weakening it, so more charge fits for the same voltage.

Charging a capacitor takes work, because each new bit of charge is pushed against the charge already there. The voltage rises steadily from 0 as charge builds up, so the average voltage is half the final one. The stored energy is the area under the charge–voltage graph:

U = ½QV = ½CV²

Doubling the voltage stores four times the energy. That's how a camera flash works: a battery slowly charges a capacitor to a few hundred volts, and the flash tube then releases that energy in a fraction of a millisecond. A defibrillator stores up to about 200–360 J and delivers it across a patient's chest in a few milliseconds.

Capacitors combine the opposite way to resistors.

• In parallel, each capacitor gets the full voltage and the charges add, so C = C₁ + C₂ + …. It's like making the plates bigger.
• In series, each capacitor holds the same charge and the voltages add, so 1/C = 1/C₁ + 1/C₂ + …. It's like making the gap wider, and the total is less than the smallest one.

A capacitor charged through a resistor doesn't fill instantly. At first current flows easily, but as the capacitor's voltage rises it pushes back, so the current shrinks and charging slows down. The time scale is the time constant:

τ = RC

with R in ohms and C in farads giving seconds. After one time constant, a charging capacitor reaches about 63% of the supply voltage; a discharging one falls to about 37% of where it started. After 5τ it's within 1% of the end, and we treat it as full or empty. When discharging, V = V₀e^−t/RC.

Time constants are everywhere: they set how long a flash takes to recharge, keep blinking lights and timers ticking, and smooth bumpy voltages in power supplies. A touchscreen works because your finger, a conductor, changes the capacitance at the spot you touch, and the phone senses the tiny change in how fast that spot charges.

The capacitor is older than the battery. Long before anyone could make a steady current, people were storing electric charge in glass jars, and sometimes getting a nasty shock from them.