On the Conservative Nature of
Electrostatic Fields
1.1 Electrostatics
When charges are permanently fixed in space the electric field produced does not
vary with time and it is therefore called static.
For two point charges (i.e. charges that occupy a region of space much smaller than
the distance between them) at rest, the electric force exerted on a charge q’ due to a
charge q is proportional to the product of the two charges and inversely proportional
to the square of the distance between them. The electric force exerted has been
experimentally derived by Coulomb and is expressed by the following equation:
where ûr is the unit vector in the direction that goes from q to q’, and R is the
distance between the two charges. If the two charges are at rest in the free space the
constant of proportionality is equal to:
with c being the speed of light, and ε0 the permittivity of the free space. The speed
of light is about equal to:
and the permittivity of free space is about equal to:
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If the two charges are in a different medium the constant of proportionality is equal
to:
where ε is the permittivity of the medium in which the charges are immersed, ε0 is
the permittivity of free space, and εr is called the relative permittivity of the
medium.
It is then apparent from Coulomb’s law that the electric field at a point P produced
by a point charge q’ at rest is given by:
where
is the distance between the location of the source charge q’ and the
observation point P and ûr is the unit vector in the direction that goes from the
location of q’ to P.
When a large number of charges are present since each charge exerts a force on all
others, we expect that at any point in space the “effect” due to a set of point charges
is equal to the sum of the effect due to the individual charges (principle of
superposition). The electric field produced by a set of N point charges at a point P is
given by:
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where qk is the magnitude of the k-th charge,
is the distance between the
location of the k-th charge and the observation point P, ûk is the unit vector in the
direction that goes from the location of the k-th charge to the point P.
Often it is convenient to ignore the fact that charge is quantized and think of it as
being distributed in a continuous way. This is reasonable as far as we are not
interested in what is happening on too small a scale. Continuous charge
distributions can be conveniently described as charge densities ρ. If the amount of
charge distributed in a small volume Δv’ located at P’ is Δq’ then the volume charge
density ρv is defined by:
The differential electric field at a point P due to a differential amount of charge
dq’=ρv⋅dv’ contained in a differential volume dv’ is:
and the electric field at point P due to a charge distributed over a volume of space
V’ is given by:
Similarly, if the charge is distributed on surface S’ with charge density ρs then:
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and the electric field at point P due to a charge distributed over a surface S’ with
density ρs is given by:
Finally, if the charge is distributed over a line L’ with density ρl then:
and the electric field at point P due to a charge distributed over a line L’ with
density ρl is given by:
It is apparent from the above equations that if the positions of all the charges (either
discrete or continuous) are specified finding the electric field simply involves
computing the appropriate variation of Coulomb’s law. Unfortunately, for most
problems of practical interest the distribution of the charges is not known so the
electric field cannot be computed using Coulomb’s law. In all cases where direct
use of Coulomb’s law is not feasible, Gauss’ law provides a convenient method to
express the inverse square action in term of the total charge rather than the specific
position of charges. Gauss’ law is just an alternative expression, in a different form
of Coulomb’s law. It does not provide any additional information about the way
static charges interact. Gauss’ law state that the “electric flux” emanated through an
arbitrary closed surface surrounding charges is:
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where Qint is the total net charge inside the closed surface S, dS is one of the small
surface element that forms the closed surface S, and ûN is the unit vector normal to
the surface element dS oriented in the outward direction.
Figure. Gauss's law states that the outward flux of D through a surface is
proportional to the enclosed charge Q.
For a discrete distribution of charges:
For a continuous distribution of charges:
where V is the volume enclosed by the closed gaussian surface S, and ρv is the
volume density of electric charge inside the volume V.
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Derivation of Gauss’ law
Gauss’ law can be derived by computing the electric flux of a single point charge
through a spherical surface, and then generalizing the result to any charge
distribution using the superposition principle.
The electric field of a single point charge at a distance R is:
The electric flux through a sphere of ray R is:
Once we know how to produce and compute electrostatic fields it is useful to realize
that if we wish to carry a negligibly small test charge along some path from point “a”
to point “b” against the electric forces we need to do work. In other words to
assemble systems of charges we need to do work and this work can be recovered
when they are released (i.e., the system possesses potential energy).
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Thus:
In general, such an integral depends on the path we take to go from “a” to “b”. If
this is the case we can get or lose work out of the field by carrying the charge to “b”
along a path and then back to “a” along another path. In principle there is nothing
wrong about “extracting” or “losing” work from a field of forces. Energy can be
extracted or dissipated from the field if the motion of the test charge produces
forces that can influence the field, for example moving the charges that produce the
field. In electrostatic, however, we assume that the charges producing the field are
fixed in position, so the work done against an electrostatic field must be conserved
(i.e., we cannot lose or get work out of the field), which in turn implies that the
work done is independent of the path. Since the forces responsible for the
electrostatic field are the Coulomb’s forces, independence from the path can be
proved taking the electric field produced by a single point charge, computing the
work done carrying an infinitesimally small test charge from a to b along different
paths and then extending the result to any charge distribution by using the
superposition principle:
Every time we move the test charge along the circular part (i.e., perpendicular to the
field) we do no work.
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Since the work done depends only on the initial and final position of the test charge,
the result of the integral can be represented as the difference between two numbers.
We call the above difference the electrostatic potential difference (or voltage
difference):
V(P) represents the amount of energy potentially available when a positive test
charge is at point P assuming that the test charge has been carried to the point P from
infinitely far away.
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Since the electrostatic potential difference depends only on the endpoints of the
integral it is easy to realize that the work done in moving a charge around a closed
path is always zero.
Such an integral is called a circulation, so a common way of expressing the
conservative nature of a field is to say that the field has no circulation or that it is
irrotational.
Figure. In electrostatics, the potential difference between P2 and P1 is the same
irrespective of the path used for calculating the line integral of the electric field
between them.: