Conductors, Capacitance and Dielectrics

Press the shutter on a camera and a flash fires — a burst of light so bright and so brief it freezes a splash of water in mid-air. The battery in that camera could never dump its energy fast enough to do that. Something else does: a small component that spends a second or two quietly hoarding charge and then unloads it all in a few thousandths of a second. That component is a capacitor, and once you start looking you find them everywhere — smoothing the ripple in every power supply, sensing your finger on a touchscreen, tuning a radio to one station, holding a bit in a memory chip, and increasingly storing serious energy in the "supercapacitors" that catch a bus's braking energy and shove it back out at the next stop.

This page is about one idea and its consequences: put equal and opposite charge on two conductors and you store energy in the field between them. To get there we first have to understand what a conductor does with charge when left to settle, define the single number — capacitance — that captures how much charge a given pair of conductors holds per volt, and then discover the cheap trick, a slab of dielectric, that makes any capacitor hold far more. Everything rests on the electric field and the potential difference it produces.

Conductors in electrostatic equilibrium

Start with a lump of metal — copper, say — carrying some charge, and leave it alone. A conductor is full of electrons free to roam. As long as there is any electric field inside the metal, those free charges feel a force and keep moving. They only stop when the field they themselves create exactly cancels everything, so that the total field inside the conductor drops to zero. That settled, currentless state is called electrostatic equilibrium, and it is reached in a matter of nanoseconds. From it, four facts follow — and they are the whole reason capacitors work.

The field inside is zero. \vec{E} = 0 everywhere within the conducting material — if it weren't, charges would still be moving and it wouldn't be equilibrium.
All net charge sits on the surface. Any excess charge is pushed as far apart as it can get and ends up spread over the outer surface; the interior stays neutral.
Just outside, the field is perpendicular to the surface and has magnitude E = \sigma/\varepsilon_0, where \sigma is the local surface charge density. Any sideways component would drive charge along the surface until it vanished.
The whole conductor is an equipotential. With \vec{E}=0 inside, no work is needed to move a charge from one point of the conductor to another, so every point sits at the same potential V — the surface included.

There is a bonus. Hollow out a cavity inside the conductor (with no charge in the cavity) and the field there is also zero — the metal shell shields its interior completely. That is the Faraday cage, and it is why the charge on a conductor gives us such clean control: put +Q on one plate and -Q on another, and the field lives entirely in the gap between them.

Capacitance: charge per volt

Take two conductors, give one a charge +Q and the other -Q. A potential difference V appears between them, and — here is the key experimental fact — doubling the charge doubles the voltage. The ratio stays fixed. That fixed ratio is the capacitance:

C = \frac{Q}{V} \qquad\Longleftrightarrow\qquad Q = C\,V.

Capacitance measures how much charge the pair holds for each volt across it. Its unit is the farad (\text{F}), and one farad is one coulomb per volt: 1\ \text{F} = 1\ \text{C/V}. A farad is enormous — real capacitors are usually microfarads (\mu\text{F}), nanofarads (\text{nF}) or picofarads (\text{pF}).

The subtle, beautiful point: C does not depend on Q or V at all. If you push more charge on, the voltage rises in step and the ratio is unchanged. Capacitance is fixed entirely by the geometry of the conductors — their size, shape and spacing — plus whatever material fills the gap. Change the geometry and you change C; change the charge and you don't.

The parallel-plate capacitor, derived

The simplest capacitor is two flat plates of area A a small distance d apart. We can work out its capacitance from scratch using the two conductor facts above. Put charge +Q on one plate and -Q on the other. The charge spreads over the facing surfaces with density \sigma = Q/A, and just outside a conductor the field is E = \sigma/\varepsilon_0. Between the plates the two contributions add to a uniform field:

E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A}.

Because the field is uniform, the voltage across the gap is simply field times distance, V = E\,d:

V = E\,d = \frac{Q\,d}{\varepsilon_0 A}.

Now form C = Q/V — the Q cancels, exactly as promised, leaving pure geometry:

C = \frac{\varepsilon_0 A}{d}.
Bigger plates hold more (C \propto A); closer plates hold more (C \propto 1/d) — squeezing the gap raises the field per volt and lets more charge crowd on.
The constant \varepsilon_0 \approx 8.85\times 10^{-12}\ \text{F/m} is the permittivity of free space.

Flat plates are just the easy case. Roll the plates into concentric cylinders and you get a cylindrical capacitor (the guts of every coaxial cable); nest two spheres and you get a spherical one. Each has its own geometry formula, but the story is identical: charge cancels, and C is set by shape and size alone.

Energy stored in a capacitor

Charging a capacitor is work. To move the next little parcel of charge dq onto a plate already at voltage v = q/C costs dW = v\,dq = (q/C)\,dq. Add up all the parcels from empty to the final charge Q (an integral of a straight line, so just the triangle's area) and the total stored energy is

U = \int_0^Q \frac{q}{C}\,dq = \frac{Q^2}{2C}.

Using Q = CV this same energy can be written three equivalent ways — pick whichever matches the quantities you know:

U = \tfrac{1}{2}\,C V^2 = \tfrac{1}{2}\,\frac{Q^2}{C} = \tfrac{1}{2}\,Q V.
The energy really lives in the field. Per unit volume of gap it is the energy density u = \tfrac{1}{2}\,\varepsilon_0 E^2 — stronger field, more energy packed in.

Notice the square: because U = \tfrac{1}{2}C V^2, doubling the voltage quadruples the stored energy. Drag the slider below to watch the stored energy climb as a parabola against voltage, and see how a larger capacitor lifts the whole curve.

Dielectrics: filling the gap

Slide a slab of insulator — glass, plastic, ceramic, even paper — into the gap and the capacitance goes up, often by a factor of tens or hundreds. Such an insulator used this way is a dielectric, and the factor is its dielectric constant \kappa (kappa), a pure number greater than 1.

Why does it help? The dielectric's molecules are little dipoles (or become so in a field). The capacitor's field polarizes them — they line up, positive ends toward the negative plate — and this alignment creates a field of its own that opposes the original. The net field inside the dielectric is weakened:

E = \frac{E_0}{\kappa}.

A weaker field means a smaller voltage V = Ed for the same charge — and since C = Q/V, a smaller V means a larger C. Every parallel-plate formula simply picks up a factor \kappa:

C = \kappa\,\frac{\varepsilon_0 A}{d} = \kappa\,C_0.
The field inside is cut to E = E_0/\kappa by the polarization of the material.
A solid dielectric also withstands a far stronger field before sparking through — its dielectric strength — so the capacitor can be run at a higher voltage as well as holding more charge.

Typical values: air is essentially \kappa = 1, paper about 3.7, mica 6, and some ceramics reach into the thousands. That is how a fingernail-sized ceramic chip can pack in microfarads.

Combining capacitors: series and parallel

Wire two capacitors together and they behave as a single equivalent capacitor — but the rule flips depending on how you connect them, and here lies the classic trap.

In parallel, both capacitors share the same voltage V, and the charges simply add (Q = Q_1 + Q_2). So the capacitances add directly — like putting the plates side by side to make one bigger plate:

C_\text{parallel} = C_1 + C_2 + \cdots

In series, the same charge Q sits on every capacitor while the voltages add (V = V_1 + V_2). Now it is the reciprocals that add — like widening the total gap, which reduces the capacitance below even the smallest member:

\frac{1}{C_\text{series}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots

For just two in series this rearranges to the handy product-over-sum form C_\text{series} = \dfrac{C_1 C_2}{C_1 + C_2}. If this feels backwards, that is because it is: it is exactly opposite to how resistors combine, where series resistances add and parallel ones take reciprocals.

Worked examples

Example 1 — capacitance from geometry. Two square plates of area A = 0.20\ \text{m}^2 are held d = 1.0\ \text{mm} = 1.0\times 10^{-3}\ \text{m} apart in air. Then

C = \frac{\varepsilon_0 A}{d} = \frac{(8.85\times 10^{-12})(0.20)}{1.0\times 10^{-3}} \approx 1.77\times 10^{-9}\ \text{F} = 1.77\ \text{nF}.

Example 2 — charge and energy at a voltage. Connect that C = 1.77\ \text{nF} capacitor across V = 12\ \text{V}. The stored charge is

Q = C V = (1.77\times 10^{-9})(12) \approx 2.1\times 10^{-8}\ \text{C} = 21\ \text{nC},

and the stored energy is

U = \tfrac{1}{2} C V^2 = \tfrac{1}{2}(1.77\times 10^{-9})(12)^2 \approx 1.3\times 10^{-7}\ \text{J} = 0.13\ \mu\text{J}.

Example 3 — inserting a dielectric. Slide a sheet of mica (\kappa = 6.0) between the same plates, keeping the charge fixed. The capacitance jumps to

C = \kappa\,C_0 = 6.0 \times 1.77\ \text{nF} \approx 10.6\ \text{nF},

the field (and so the voltage) drops to one sixth, and the plates now hold six times the charge at the same voltage.

Example 4 — a series-and-parallel combo. Take C_1 = 2\ \mu\text{F} in parallel with C_2 = 4\ \mu\text{F}, and that pair in series with C_3 = 3\ \mu\text{F}. First the parallel pair adds directly:

C_{12} = C_1 + C_2 = 2 + 4 = 6\ \mu\text{F}.

Then combine C_{12} in series with C_3 by product-over-sum:

C_\text{total} = \frac{C_{12}\,C_3}{C_{12} + C_3} = \frac{6 \times 3}{6 + 3} = \frac{18}{9} = 2\ \mu\text{F}.

Watch out — this is the mix-up that costs marks in every exam. The rules for capacitors are the mirror image of the rules for resistors:

Capacitors in parallel add directly (C = C_1 + C_2); in series they take reciprocals.
Resistors in series add directly (R = R_1 + R_2); in parallel they take reciprocals.

The way to never forget it: think about what "parallel" does physically. Parallel capacitors are like bolting the plates side by side — more plate area, so more capacitance, so they add. Parallel resistors are like opening extra lanes for the current — more paths, less resistance, so the total drops. A quick sanity check whenever you finish a calculation: a series combination of capacitors must come out smaller than the smallest capacitor in it, and a parallel combination must come out larger than the largest. If your answer breaks that rule, you've swapped the formulas.

Not because of the rubber tyres — a bolt that has just jumped a kilometre of air is not going to be stopped by a few centimetres of rubber. You are safe because the car's metal body is a conductor in electrostatic equilibrium: a Faraday cage. When the charge from the strike arrives, it races to the outer surface (net charge on a conductor lives on the surface), and the field inside the shell stays zero. The current flows around you, over the skin of the car, and down to the ground, while the cabin — the cavity — is shielded.

Michael Faraday demonstrated this in 1836 by sitting inside a foil-lined cage while sparks crackled all over the outside; his electroscopes inside never budged. The same principle protects the sensitive chips inside your phone (metal casing), keeps a microwave oven's radiation in (the mesh in the door is a cage with holes smaller than the wavelength), and, three centuries earlier, first stored electric charge in the Leyden jar of 1745 — a water-filled glass jar, the very first capacitor, whose inner and outer foils were two conductors and whose glass was the dielectric.