What is a Capacitor: Definition, Summary & Equations

[Greg Bernhardt]

Definition/Summary

A capacitor (or condenser) is something across which charge cannot move, but across which an accumulation of charge on one side can affect charge on the opposite side.

So a capacitor does not conduct direct (DC) current, but does conduct alternating (AC) current.

The two sides are usually described as plates, and the material between as the dielectric.

When a steady potential difference (voltage) is first applied across a capacitor, equal positive and negative charge will build up on its two plates, until a maximum is reached.

The capacitance, $C$, of the capacitor is the ratio of that maximum charge to the applied voltage: $C\,=\,Q/V$. It is measured in farads (coulombs per volt).

A capacitor is a storage device for charge, since any charge difference between its plates will remain there so long as no electric path joins the plates externally. There is no loss of energy if it is charged or discharged through an inductor (but up to fifty percent loss through a resistor).

Equations

Charge across a capacitor $C$:

$$Q = VC,$$

where $V$ is the potential difference.

Across two capacitors $C_1$ and $C_2$ in parallel:

$$Q = VC_1 + VC_2 = V(C_1\,+\,C_2)$$

Across two capacitors $C_1$ and $C_2$ in series:

$$Q = V_1C_1 = V_2C_2 = \frac{(V_2C_2)C_1 + (V_1C_1)C_2}{C_1 + C_2} = (V_1\,+\,V_2)\frac{C_1C_2}{C_1 + C_2} = (V_1\,+\,V_2)\left(\frac{1}{\frac{1}{C_1}\,+\,\frac{1}{C_2}}\right)$$

Energy stored:

$$\frac{1}{2}\,C\,V^2$$

Extended explanation

Displacement current:

No current ever flows through a functioning capacitor.

But while a capacitor is charging or discharging (that is, neither at zero nor maximum charge), current is flowing round the circuit joining the plates externally, and so there would be a breach of Kirchhoff's first rule (current in = current out at any point) at each plate, if only ordinary current were used, since there is ordinary current in the circuit on one side of the plate, but not in the dielectric on the other side.

Accordingly, a displacement current is deemed to flow through the capacitor, restoring the validity of Kirchhoff's first rule:

$$I\ =\ C\frac{dV}{dt}$$

and this linear displacement current $I$ (which might better be called the flux current or free flux current) is the rate of change of the flux (field strength times area) of the electric displacement field $D$:

$$I\ =\ A\,\widehat{\mathbf{n}}\cdot\frac{\partial\mathbf{D}}{\partial t}\ =\ A\,\frac{\partial D}{\partial t}\ =\ C\frac{dV}{dt}$$

which appears in the Ampére-Maxwell law (one of Maxwell's equations in the free version):

$$\nabla\,\times\,\mathbf{H}\ =\ \mathbf{J}_f\ +\ \frac{\partial\mathbf{D}}{\partial t}$$

Note that the displacement alluded to in the displacement current across a capacitor is of free charge, and is non-local, since it alludes to charge being displaced from one plate to the other, which is a substantial distance compared with the local displacement of bound charge in, for example, the presence of a polarisation field.

Impedance to steady AC current of frequency ω:

A capacitor has no resistance: there is no direct relation between $V\text{ and }I$.

Instead, a capacitor has reactance: there is a direct relation between $dV/dt\text{ and }I$ …

$dV/dt = (1/C)dQ/dt = CI$​

… the current leads the voltage by 90º ("half-out-of-phase").

The convention $dV/dt = j\omega V$ gives the impedance:

$Z = V/I = 1/j\omega C$​

By contrast, an inductor also has no resistance, and an impedance defined by $Z = V/I = (LdI/dt)/I = (j\omega LI)/I = j\omega L$

Across capacitor of plate area A and plate distance d (approximate, for small d/A):

(All displacements fields and potential differences are measured in the direction from the -ve to the +ve plate)

(Formulas for a cylindrical capacitor are in blue, and for a spherical capacitor are in red)

Electric displacement field (charge/area):

$D\ =\ \frac{-Q}{A}\ \$$\frac{-Q}{2\pi rh}\ \$$\frac{-Q}{4\pi r^2}$


Electric field (force/charge):

$E\ =\ \frac{D}{\varepsilon}\ =\ \frac{-Q}{\varepsilon\,A}\ \$$\frac{-Q}{\varepsilon\,2\pi rh}\ \$$\frac{-Q}{\varepsilon\,4\pi r^2}$

Voltage:

$V\ =\ -\int_{d_1}^{d_2} E\,dx\ =\ \frac{d\,Q}{\varepsilon\,A}\ \$$\frac{(ln(r_2)-ln(r_1))\,Q}{\varepsilon\,\,2\pi h}\ \$$\frac{(1/r_1-1/r_2)\,Q}{\varepsilon\,\,4\pi}$

Capacitance = charge/voltage:

$C\ =\ \frac{Q}{V}\ =\ \frac{\varepsilon\,A}{d}\ \$$\frac{\varepsilon\,2\pi h}{ln(r_2/r_1)}\ \$$\frac{\varepsilon\,4\pi r_1r_2}{(r_2-r_1)}=\frac{\varepsilon\,\sqrt{A_1A_2}}{d}$

Energy stored, in capacitor of any shape:

(= work done in increasing charge from $0\text{ to }Q$ with fixed plates):

$$W\ =\ \int \text{charge.voltage}\ =\ \int_0^Q V\,dq\ =\ \int_0^Q \,\frac{q}{C}\,dq\ =\ \frac{Q^2}{2C}\ =\ \frac{1}{2}\,C\,V^2\ =\ \frac{1}{2}\,Q\,V$$

Force per charge on either plate:

Electric force per charge is usually the Lorentz force per charge, equal to the electric field, $\mathbf{E}$.

However, this applies only to a "test" charge (one small enough not to affect the field), and not to a charge which causes a discontinuity in the field, and "drags" that discontinuity with it when it moves (in a capacitor, one plate interrupts the field of the other plate, causing it to disappear on the far side).

Instead, the force per charge on each plate is $(1/Q)\mathbf{\nabla}W = \mathbf{\nabla}V/2 = \mathbf{E}/2$: in other words, the force per charge on each plate is half the usual Lorentz force.

Note that (except in the parallel plate case) the force per charge on each plate is different: Newton's third law applies only to total force, which of course in the cylindrical or spherical case is zero.

For example, the forces per charge on the plates of a spherical capacitor are $Q^2/8\varepsilon\pi r_1^2$ outward and $Q^2/8\varepsilon\pi r_2^2$ inward: the former is entirely the self-repulsive force of the inner sphere, with no force from the surrounding outer sphere: and the latter is equal and opposite to the self-repulsive force of the outer sphere, with double that force inward from the inner sphere.

Capacitor v. insulator:

Similarity: charge cannot move across a capacitor or an insulator.

Difference: an accumulation of charge on one side can affect charge on the opposite side of a capacitor, but not of an insulator.

An insulator (or a switch) may be considered a capacitor with zero capacitance, and therefore with zero charge and zero energy at any voltage.

Inverse exponential rate of charging:

A capacitor does not charge or discharge instantly.

When a steady voltage $V_1$ is first applied, through a circuit of resistance $R$, to a capacitor across which there is already a voltage $V_0$, both the charging current $I$ in the circuit and the voltage difference $V_1\,-\,V$ change exponentially, with a parameter $-1/CR$:

$$I(t) = \frac{V_1\,-\,V_0}{R}\,e^{-\frac{1}{CR}\,t}$$

$$V_1\ -\ V(t) = (V_1\,-\,V_0)\,e^{-\frac{1}{CR}\,t}$$

So the current becomes effectively zero, and the voltage across the capacitor becomes effectively $V_1$, after a time proportional to $CR$.

Energy loss:

Energy lost (to heat in the resistor):

$$\int\,I^2(t)\,R\,dt\ =\ \frac{1}{2}\,C (V_1\,-\,V_0)^2$$

Efficiency (energy lost per total energy):

$$\frac{V_1^2\,-\,V_0^2}{V_1^2\,-\,V_0^2\,+\,(V_1\,-\,V_0)^2}\ =\ \frac{1}{2}\,\left(1\,+\,\frac{V_0}{V_1}\right)$$

Accordingly, charging a capacitor through a resistor is very inefficient unless the applied voltage stays close to the voltage across the capacitor.

But there is no energy loss on charging a capacitor through an inductor, basically because the applied voltage then appears across the inductor instead of across the capacitor.

Fluid mechanics analogy:

In fluid mechanics, a capacitor is analogous to a diaphragm blocking a pipe.

Water cannot pass through the diaphragm.

But pressure suddenly applied on one side will make the diaphragm bulge, so that water continues to flow "into" the diaphragm on one side, and "away from" the diaphragm on the other side, until the diaphragm reaches its maximum bulge for that pressure.

The volume of the bulge, divided by the pressure, is the capacitance.

The energy stored is half the volume of the bulge times the pressure.

The displacement current is rate at which the volume of the bulge is increasing (and is zero once the maximum bulge is reached).

Polarised capacitor (or polar capacitor):

This has one metal and one electrolyte plate (instead of two metal plates), and its dielectric is the oxide of the metal.

It has nothing to do with polarisation current.

It simply means that it behaves as a capacitor in one direction only (with the metal plate at the positive potential), and as a conductor in the other direction.

So it only works one way round in a DC circuit, and behaves as a rectifier in an AC circuit.

Dielectric strength:

The dielectric strength of a capacitor is the voltage at which a spark will jump across the dielectric (from one plate to the other).

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[Aaron Paulo]

This is a great explanation of what a capacitor is and how it works. Thanks for taking the time to put this together!