Capacitance and capacitors

A capacitor is a device that stores charge and the energy of an electric field. You find one in a camera flash, a defibrillator, the tuned circuit of a radio, computer memory, motor-starting circuits and on almost every circuit board. This topic applies what we know about electric fields to practical problems and prepares the ground for “Electromagnetic oscillations”.

§1. Capacitance

When an isolated conductor is given a charge, its potential rises in proportion to the charge. The constant of proportionality shows how well the conductor stores charge:

Capacitance does not depend on the charge or the voltage, only on the size, shape and arrangement of the conductors and on the surrounding medium. Doubling the charge doubles the voltage, and the ratio stays the same.

1 F is a huge unit: a sphere with capacitance 1 F would need a radius of 9·10⁹ m (about 1400 times the Earth’s radius). In practice smaller units are used: 1 μF = 10⁻⁶ F, 1 nF = 10⁻⁹ F, 1 pF = 10⁻¹² F.

§2. How a capacitor is built; the parallel-plate capacitor

A single conductor has a small capacitance. To store a lot of charge we use a capacitor: two conductors (plates) separated by a thin layer of dielectric. When charged, the plates carry equal and opposite charges (+q and −q); the charge of a capacitor means the charge q on one plate. The field is almost entirely between the plates.

A parallel-plate capacitor is two parallel plates a distance d apart:

• Larger plate area S: larger capacitance.
• Larger separation d: smaller capacitance.
• Inserting a dielectric raises the capacitance ε times (the dielectric polarises and weakens the field, so more charge fits at the same voltage).

The field inside is uniform: E = U/d.
By dielectric, capacitors may be paper, ceramic, mica or electrolytic; in a variable capacitor one set of plates rotates and changes the overlapping area, which is how a radio is tuned. Every capacitor has a maximum working voltage; above it the dielectric breaks down.

Worked example

Each plate has an area of 100 cm², the gap is 1 mm and filled with air. Find the capacitance.

1. Convert to SI: S = 100·10⁻⁴ = 10⁻² m², d = 10⁻³ m.
2. C = ε₀S/d = 8.85·10⁻¹²·10⁻²/10⁻³.
3. C = 8.85·10⁻¹¹ F ≈ 88.5 pF.

§3. Connected to a source or disconnected

If a problem changes the capacitor’s parameters (d, S, ε), first decide its situation:

• Still connected to the source: the voltage is fixed, U = const. If the capacitance rises, extra charge flows from the source: q = CU.
• Disconnected: the charge has nowhere to go, q = const. If the capacitance rises, the voltage falls: U = q/C.

This also decides how the energy and the field strength change.

Worked example

The plate separation of a disconnected capacitor is doubled. How do C, U, E and W change?

1. q = const; C = εε₀S/d halves.
2. U = q/C doubles.
3. E = U/d stays the same (both U and d doubled).
4. W = q²/(2C) doubles: an external force did work pulling the plates apart.

§4. Combining capacitors

• Two capacitors in series: C = C₁C₂/(C₁ + C₂).
• n identical capacitors: parallel nC₀, series C₀/n.
• The formulas are the reverse of those for resistors: parallel connection effectively increases the plate area, series connection effectively increases the gap.

In series the voltage divides in inverse proportion to the capacitances: the smaller capacitor gets the larger voltage. So capacitors are put in series to withstand a high voltage and in parallel to get a large capacitance.

Worked example

A dielectric of permittivity ε completely fills an air capacitor of capacitance C, and a capacitor Cₓ is connected in series with it; the total capacitance is C/2. Find Cₓ.

1. Capacitance with the dielectric: εC.
2. Series: 2/C = 1/(εC) + 1/Cₓ → 1/Cₓ = (2ε − 1)/(εC).
3. Cₓ = εC/(2ε − 1).

§5. Energy stored in a capacitor

While charging, the source moves charge from one plate to the other and does work; since the voltage rises gradually from zero to U, the average voltage is U/2:

The energy is stored in the electric field between the plates. Pulling apart the plates of a disconnected capacitor requires work, so the energy grows; for a connected capacitor the source exchanges energy as well.

Putting C = εε₀S/d and U = Ed into W = CU²/2 for a parallel-plate capacitor gives the energy per unit volume:

This formula holds for any electric field, not only in a capacitor: the energy is in the field itself, not in the charges.

When a charged capacitor is connected to an uncharged one, charge is conserved, but some energy is lost as heat in the wires and as radiation.

Worked example

A 10 μF capacitor is charged to 200 V. Find the charge and the energy.

1. q = CU = 10⁻⁵·200 = 2·10⁻³ C.
2. W = CU²/2 = 10⁻⁵·4·10⁴/2 = 0.2 J.

Worked example

A 2 μF capacitor charged to 100 V is connected in parallel with an identical uncharged one. How do the voltage and energy change?

1. Charge is conserved: q = 2·10⁻⁶·100 = 2·10⁻⁴ C; the total capacitance is 4 μF.
2. U = q/C = 2·10⁻⁴/(4·10⁻⁶) = 50 V.
3. Energy: before 2·10⁻⁶·100²/2 = 0.01 J, after 4·10⁻⁶·50²/2 = 0.005 J; half became heat.

§6. Uses of capacitors

• Delivering energy fast: a camera flash and a defibrillator charge a capacitor slowly and release the energy in a fraction of a second, giving a very large power.
• Tuned circuits: a capacitor and a coil set the frequency in radio receivers and transmitters.
• Filters: a capacitor blocks direct current but passes alternating current; in power supplies it smooths out voltage ripple.
• Touch screens and keyboards: a finger changes the capacitance, and the device detects it.
• Supercapacitors have capacitances of farads or more and serve as energy stores that charge and discharge quickly.

Check yourself

1. With a charge of 4 μC the voltage across a capacitor is 20 V. Capacitance?

2. 2 μF and 3 μF capacitors are connected in parallel. Total capacitance?

3. And if the same two are in series?

4. A dielectric with ε = 2 is inserted into a capacitor connected to a source. By what factor does the charge change?

5. The voltage triples. By what factor does the energy grow?

6. Three 6 μF capacitors are in series. Total capacitance?

7. What voltage stores 200 J in a 100 μF defibrillator capacitor?

8. Does capacitance depend on the voltage across the capacitor?

Common mistakes

• Writing combination formulas as for resistors: for capacitors they are reversed.
• Not distinguishing U = const from q = const.
• Not converting μF, pF and cm² to SI.
• Thinking capacitance depends on voltage or charge.
• Assuming energy is conserved when capacitors are connected: it is the charge that is conserved.