A single conductor holds almost nothing. Bring a second one close and each holds the other’s charge in place; together they keep what neither could alone. We learned that from two plates. Later we learned it from each other.
⟨ transmission ends ⟩This fragment teaches you to size a capacitor: to say how much charge a pair of conductors holds for every volt between them, and to place any real part on the ladder that runs from picofarads to farads. By the end you will also know how much energy a charged capacitor keeps.
Fragment 05 ended with 2 conductors brought close and kept apart by an insulator. That pair has a name: a capacitor. Below are 2 square steel plates, 15 cm on a side and 2 mm apart, with air between them. A small pump carries charge from the lower plate to the upper one, 1 nC at a press, so the upper plate goes positive and the lower one equally negative. A voltmeter (Fragment 04) reads the voltage between them.
Push twice the charge onto the same plates. What does the voltage between them do?
Plates 15 cm square, 2 mm apart in air: 100 pF. The field is taken as even between the plates and 0 outside; the real edges add about 3%. The gap is drawn 25× wider than it is so you can see into it.
Every press added the same 10 V. Charge and voltage rose in step: twice the charge, twice the voltage; 3 times, 3 times. The ratio Q ÷ V never moved, because it does not belong to the charge or to the voltage. It belongs to the plates. It is their own number, and it is called capacitance: how much charge the pair holds for every volt.
Read it 3 ways. It says the ratio of charge to voltage is fixed. At the bench: more capacitance holds more charge at the same voltage. In numbers kept simple on purpose, the bench’s own line last:
The farad is named for Michael Faraday. It is enormous: a steel ball the size of the Earth has a capacitance of 0.7 mF, and there are parts smaller than your thumb with more. So what decides a pair’s capacitance? Only its shape, its spacing and its filling. To see that, hold the voltage fixed instead of the charge: below, a battery (Fragment 04) holds 20 V across the same plates, and the gap can change.
Halve the gap, from 2 mm to 1 mm, while the battery holds 20 V. What happens to the charge on the plates?
Same plates, battery holding 20 V; only the gap moves. Q = C × V with C = ε0A ÷ d, edges ignored. The charge moving on or off the plates shows as a glow on the wires. The gap is drawn 25× wider than it is.
Half the gap, twice the charge. Here is why. Between the plates the field is even, and the voltage is that field spread over the gap: V = E × d. To hold 20 V across half the distance the field must be twice as strong, and twice the field takes twice the charge on the plates to make. Closer plates hold charge more cheaply.
The gap does not have to be air. Below, the same plates hold 20 V from the battery, and beside them waits a slab of glass, 2 mm thick, cut to fill the gap exactly. Glass is an insulator (Fragment 05): its charges can lean, but they cannot leave.
Slide the glass into the gap while the battery keeps 20 V across the plates. What happens to the charge on the plates?
Window glass, dielectric constant 7 (glasses run from 4 to 10), cut to fill the gap exactly. Over the glass the plates carry 7× the charge; the small dim grains on the glass faces are its own leaned charge, where the threads from 6 of every 7 plate grains end. The gap is drawn 25× wider than it is.
The plates took on 7 times the charge and gave none of it to the glass. What the glass did was lean: each atom turned its negative side toward the positive plate, so the 2 faces of the slab wear thin layers of opposite charge. Those layers pull against the plates’ field and would lower the voltage, so the battery pushed on more charge until the field was back to 20 V ÷ 2 mm. A filling that does this is a dielectric, and the factor it multiplies the capacitance by is its dielectric constant: about 7 for glass, 2 to 3 for the plastics in film capacitors, thousands for some ceramics.
Everything the gap and the glass did is one formula, and it has an honest descriptive name: the parallel-plate formula.
A whole square metre of plates, a millimetre apart, and not even 10 nF. Real capacitors do not look like 2 plates in air, and size is the reason. Below, a dial climbs the ladder from 1 pF to 1 F, showing at each step how big air-spaced plates 1 mm apart would have to be, and what part actually lives there.
1 µF is an everyday part, smaller than a grain of rice. Made instead from bare plates in air, 1 mm apart, how big would each plate need to be?
Plate side = √(C × 1 mm ÷ ε0), edges ignored. The family bands are the common ranges (ceramic pF to about 10 µF, film 100 pF to about 10 µF, electrolytic about 1 µF to 20 mF, supercapacitors 0.1 F to thousands of F); parts exist outside them.
Plates the size of a flat for 1 µF, the size of a city for 1 F. Real parts cheat, all in the same 2 ways: enormous area rolled or stacked into a small body, and a gap so thin that air could not hold it, since air sparks across at about 3 kV per mm (Fragment 03). How a family does it is what makes it a family:
Drawn to different scales: the chip is about 1.6 mm long, the disc and the box about 1 cm, the can about 1 cm across. Colours vary by maker; the shape and the printed value are what to trust.
One more thing the plates hold. Their charges are opposite, so the 2 plates attract each other (Fragment 01), and holding them apart takes a force. Below, the plates carry 2 nC from the 20 V battery, and then the battery is taken away, so the charge is trapped. The voltmeter stays.
The plates are disconnected, holding 2 nC. Pull them to twice the gap. What does the energy they hold do?
The plates hold 2 nC from the 20 V charge; then the battery is disconnected. V = Q ÷ C, U = ½ Q V, and the pull is Q² ÷ (2 ε0 A) = 10 µN at every gap, edges ignored. The gap is drawn 25× wider than it is.
The charge never changed. The voltage doubled, because plates twice as far apart have half the capacitance. And the energy doubled: the work of your hand against a 10 µN pull over 2 mm, 20 nJ, went into the field between the plates and waits there. That is what a charged capacitor holds besides charge: energy, and its formula is called the energy stored in a capacitor.
The half is there because the first bit of charge cost nothing to push on and the last bit cost the full voltage, so on average each coulomb cost half of V. The last line is the one to remember: 8.7 J, still there after the camera is switched off, waiting for whatever next touches both leads. Fragment 35 is about that.
A part must hold 10 µC at 5 V. What capacitance do you need?
Three parts: 100 pF, 100 nF and 1000 µF. Which line puts each one with the family that makes it?
A 1 µF capacitor is charged to 5 V. How much energy does it hold?
You can now size a capacitor from what it must hold, say where on the ladder a real part lives and which family makes it, and count the energy it keeps. The charge that filled the plates had to travel to them through a wire, and while it travelled it was something new: charge in motion, a current. That is Fragment 07.