Capacitance

Capacitance is a measure of the amount of electric charge stored (or separated) for a given electric potential. The capacitance is usually defined as the total electric charge placed on the object divided by the potential of the object:

C = \frac{Q}{V}

or, according to Gauss's law, the capacitance can be expressed as the electric flux per volt

C = \frac{\Phi}{V}

where

C is the capacitance in farads, F

Q is the charge in coulombs, C

V is the potential in volts, V

\Phi

is the electric flux associated with the charge Q in coulombs

Compare this form with the definition of inductance.

Introduction

Capacitance exists between any two conductors insulated from one another. The formula defining capacitance above is valid if it is understood that the conductors have equal but opposite charge Q, and the voltage V is the potential difference between the two conductors.

The SI unit of capacitance is the farad (F). A capacitance of one farad results in a potential of one volt for one coulomb of charge. The capacitance of the majority of capacitors used in electronic circuits is several orders of magnitude smaller than the farad. The most common units of capacitance in use today are the microfarad (µF), the nanofarad (nF) and the picofarad (pF).

It should be noted that the above equation (C = Q/V) is only applicable for values of Q which are much larger than the electron charge e = 1.602×10^-19 C. For example, if a capacitance of 1 pF is charged to a voltage of 100 nV, the equation would predict a charge Q = 10^-19 C, which is smaller than the charge on a single electron.

The capacitance can be calculated if the geometry of the conductors and the dielectric properties of the insulator between the conductors are known. For example, the capacitance of a parallel-plate capacitor constructed of two identical plane electrodes of area A at constant spacing d is approximately equal to the following:

C = \epsilon_0 \epsilon_r \frac{A}{d}

where

C is the capacitance in farads, F

ε₀ is the permittivity of free space, measured in farads per meter

ε_r is the dielectric constant or relative permittivity of the insulator used

A is the area of each plane electrode, measured in square metres

d is the separation between the electrodes, measured in metres

Energy

The energy (measured in joules) stored in a capacitance is equal to the work done to charge it. Consider a capacitance C, holding a charge +q on one plate and -q on the other. Moving a small element of charge dq from one plate to the other against the potential difference V = q/C requires the work dW:

dW = \frac{q}{C}dq

where

W is the work measured in joules

q is the charge measured in coulombs

C is the capacitance, measured in farads

We can find the energy stored in a capacitance by integrating this equation. Starting with an uncharged capacitance (q=0) and moving charge from one plate to the other until the plates have charge +Q and -Q requires the work W:

W_{charging} = \int_{0}^{Q} \frac{q}{C} dq = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}CV^2 = W_{stored}

Combining this with the above equation for the capacitance of a flat-plate capacitor, we get:

W_{stored} = \frac{1}{2} \epsilon_0 \epsilon_r \frac{A}{d} V^2

Capacitance and 'displacement current'

The physicist James Clerk Maxwell invented the concept of displacement current, dD/dt, to make Ampere's law consistent with conservation of charge in cases where charge is accumulating, for example in a capacitor. He interpreted this as a real motion of charges, even in vacuum, where he supposed that it corresponded to motion of dipole charges in the ether. Although this interpretation has been abandoned, Maxwell's correction to Ampere's law remains valid (a changing electric field produces a magnetic field).

Capacitance/inductance duality

In mathematical terms, the ideal capacitance can be considered as an inverse of the ideal inductance, because the voltage-current equations of the two phenomena can be transformed into one another by exchanging the voltage and current terms.

Self-capacitance

In electrical circuits, the term capacitance is usually a shorthand for the mutual capacitance between two adjacent conductors, such as the two plates of a capacitor. There also exists a property called self-capacitance, which is the amount of electrical charge that must be added to an isolated conductor to raise its electrical potential by one volt. The reference point for this potential is a theoretical hollow conducting sphere, of infinite radius, centred on the conductor. Using this method, the self-capacitance of a conducting sphere of radius R is given by:

C=4\pi\epsilon_0R

Typical values of self-capacitance are:

for the top electrode of a van de Graaf generator, typically a sphere 20 cm in diameter: 20 pF
the planet Earth: about 710 µF

Elastance

The inverse of capacitance is called elastance, and its unit is the reciprocal farad, also informally called the daraf.

Stray capacitance

Any two adjacent conductors can be considered as a capacitor, although the capacitance will be small unless the conductors are close together or long. This (unwanted) effect is termed "stray capacitance". Stray capacitance can allow signals to leak between circuits, and is a limiting factor for correct functioning of circuits at high frequency.

References

Tipler, Paul (1998). Physics for Scientists and Engineers: Vol. 2: Electricity and Magnetism, Light (4th ed.). W. H. Freeman. ISBN 1572594926
Serway, Raymond; Jewett, John (2003). Physics for Scientists and Engineers (6 ed.). Brooks Cole. ISBN 0534408427

Gourt Home::Gourt :: Capacitance

Introduction

Energy

Capacitance and 'displacement current'

Capacitance/inductance duality

Self-capacitance

Elastance

Stray capacitance

References

See also