Capacitance of a capacitor

Capacitance is a property of a conductor that characterises its ability to store electric charge. Capacitance is defined as the ratio of the charge of the conductor to its potential. Capacitance is denoted by C.

$$ Capacitance\ C = {Charge\ Q\ stored\ in\ the\ capacitor \over Potential\ difference\ V\ across\ the\ capacitor}$$

The same in English

$$ capacity\ C = {charge\ Q\ stored\ on\ conductor\ \over potential\ V\ to\ which\ conductor\ is\ raised}$$

$$ Q = {C \cdot V}$$

Let us look at capacitance from another side.

Hence we already know that

$$ work\ done = Q\cdot V = I\cdot t\cdot V$$

$$ Q = {work\ done\ \over V}$$

If you have an idea to combine these formulas as follows

$$ {work\ done\ \over V} = Q = C \cdot V\ \color{red} ^*$$

$$ {work\ done\ \over V} = C \cdot V\ \color{red} ^*$$

And then get something like

$$ C = {work\ done\ \over V^2}\ \color{red} ^*$$

$$ work\ done = C \cdot V^2\ \color{red} ^*$$

* - that is wrong, it is given as a basis for thought

Then you have to remember that although these transformations give capacitance equal to the work done divided by the square of the voltage, that is not so.

The fact was not taken into account that as soon as the capacitor starts discharging the potential difference between the plates will fall and the amount of energy will also change. Therefore in the simplest case you have to take not V but the average voltage between 0 and V, that is V/2

$$ work\ done = C \cdot V \cdot {V \over 2} = C \cdot {V^2 \over 2}$$

At the same time the well-known formula for the energy stored in a capacitor looks as follows.

$$ W = {Q^2 \over 2 \cdot C} = {Q \cdot U \over 2} = {C\ \cdot U^2 \over 2}$$

Where W is the energy and U is the potential difference. The formula coincides with the one we obtained above.

It can be derived from the following equations

$$ dA = U \cdot dq = {1 \over C} \cdot q \cdot dq $$

$$ W = {1 \over C} \cdot {\int_{0}^{Q} \cdot q \cdot dq} = {Q^2 \over 2 \cdot C} $$

That is

$$ C = {W\ \over 2 \cdot U^2}$$

The unit of capacitance is the farad

$$ farad = 1\ F = {1\ coulomb \over volt} = 1\ C\ /\ V$$

Semiconductors

Superconductivity

Applied optics

Microelectronics

Optoelectronics

Materials science

Optical spectroscopy

Antennas

Article author: Andrei Olegovich