In
electrostatics, the
coefficients
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
of potential determine the relationship between the
charge
Charge or charged may refer to:
Arts, entertainment, and media Films
* '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* ''Charge'' (David Ford album)
* ''Charge'' (Machel Montano album)
* ''Charge!!'', an album by The Aqu ...
and
electrostatic potential
Electrostatics is a branch of physics that studies electric charges at rest ( static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for ambe ...
(
electrical potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
), which is purely geometric:
:
where is the surface charge on conductor . The coefficients of potential are the coefficients . should be correctly read as the potential on the -th conductor, and hence "
" is the due to charge 1 on conductor 2.
:
Note that:
# , by symmetry, and
# is not dependent on the charge.
The physical content of the symmetry is as follows:
: if a charge on conductor brings conductor to a potential , then the same charge placed on would bring to the same potential .
In general, the coefficients is used when describing system of conductors, such as in the
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
.
Theory
System of conductors. The electrostatic potential at point is .
Given the electrical potential on a conductor surface (the
equipotential surface
In mathematics and physics, an equipotential or isopotential refers to a region in space where every point is at the same potential. This usually refers to a scalar potential (in that case it is a level set of the potential), although it can ...
or the point chosen on surface ) contained in a system of conductors :
:
where , i.e. the distance from the area-element to a particular point on conductor . is not, in general, uniformly distributed across the surface. Let us introduce the factor that describes how the actual charge density differs from the average and itself on a position on the surface of the -th conductor:
:
or
:
Then,
:
It can be shown that
is independent of the distribution
. Hence, with
:
we have
:
Example
In this example, we employ the method of coefficients of potential to determine the capacitance on a two-conductor system.
For a two-conductor system, the system of linear equations is
:
On a
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
, the charge on the two conductors is equal and opposite: . Therefore,
:
and
:
Hence,
:
Related coefficients
Note that the array of linear equations
:
can be inverted to
:
where the with are called the
coefficients of capacity
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves v ...
and the with are called the
coefficients of electrostatic induction
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves v ...
.
For a system of two spherical conductors held at the same potential,
:
If the two conductors carry equal and opposite charges,
:
The system of conductors can be shown to have similar symmetry .
References
{{Reflist
*
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and li ...
(1873)
A Treatise on Electricity and Magnetism, § 86, page 89.
Electrostatics