An electric field is generated by electrically
charged particles and time-varying magnetic
fields. The electric field describes the electric
force experienced by a motionless positively
charged test particle at any point in space
relative to the source(s) of the field. The
concept of an electric field was introduced
by Michael Faraday.
Qualitative description
The electric field is a vector field. The
field vector at a given point is defined as
the force vector per unit charge that would
be exerted on a stationary test charge at
that point. An electric field is generated
by electric charge, as well as by a time-varying
magnetic field. The electric charge can be
a single charge or a group of discrete charges
or any continuous distribution of charge.
Electric fields contain electrical energy
with energy density proportional to the square
of the field amplitude. The electric field
is to charge as gravitational acceleration
is to mass. The SI units of the field are
newtons per coulomb or, equivalently, volts
per metre, which in terms of SI base units
are kg⋅m⋅s−3⋅A−1.
An electric field that changes with time,
such as due to the motion of charged particles
producing the field, influences the local
magnetic field. That is: the electric and
magnetic fields are not separate phenomena;
what one observer perceives as an electric
field, another observer in a different frame
of reference perceives as a mixture of electric
and magnetic fields. For this reason, one
speaks of "electromagnetism" or "electromagnetic
fields". In quantum electrodynamics, disturbances
in the electromagnetic fields are called photons.
Definition
Electric Field
Consider a point charge q with position. Now
suppose the charge is subject to a force Fon
q due to other charges. Since this force varies
with the position of the charge and by Coulomb's
Law it is defined at all points in space,
Fon q is a continuous function of the charge's
position. This suggests that there is some
property of the space that causes the force
which is exerted on the charge q. This property
is called the electric field and it is defined
by
Notice that the magnitude of the electric
field has dimensions of Force/Charge. Mathematically,
the E field can be thought of as a function
that associates a vector with every point
in space. Each such vector's magnitude is
proportional to how much force a charge at
that point would "feel" if it were present
and this force would have the same direction
as the electric field vector at that point.
It is also important to note that the electric
field defined above is caused by a configuration
of other electric charges. This means that
the charge q in the equation above is not
the charge that is creating the electric field,
but rather, being acted upon by it. This definition
does not give a means of computing the electric
field caused by a group of charges.
Superposition
Array of discrete point charges
Electric fields satisfy the superposition
principle. If more than one charge is present,
the total electric field at any point is equal
to the vector sum of the separate electric
fields that each point charge would create
in the absence of the others. That is,
where Ei is the electric field created by
the i-th point charge.
At any point of interest, the total E-field
due to N point charges is simply the superposition
of the E-fields due to each point charge,
given by
where Qi is the electric charge of the i-th
point charge, the corresponding unit vector
of ri, which is the position of charge Qi
with respect to the point of interest.
Continuum of charges
It holds for an infinite number of infinitesimally
small elements of charges – i.e. a continuous
distribution of charge. By taking the limit
as N approaches infinity in the previous equation,
the electric field for a continuum of charges
can be given by the integral:
where ρ is the charge density, ε0 the permittivity
of free space, and dV is the differential
volume element. This integral is a volume
integral over the region of the charge distribution.
The equations above express the electric field
of point charges as derived from Coulomb's
law, which is a special case of Gauss's Law.
While Coulomb's law is only true for stationary
point charges, Gauss's law is true for all
charges either in static form or in motion.
Gauss's Law establishes a more fundamental
relationship between the distribution of electric
charge in space and the resulting electric
field. It is one of Maxwell's equations governing
electromagnetism.
Gauss's law allows the E-field to be calculated
in terms of a continuous distribution of charge
density. In differential form, it can be stated
as
where ∇⋅ is the divergence operator, ρ
is the total charge density, including free
and bound charge, in other words all the charge
present in the system.
Electrostatic fields
Electrostatic fields are E-fields which do
not change with time, which happens when the
charges are stationary.
The electric field E at a point r, that is,
E(r), is equal to the negative gradient of
the electric potential , a scalar field at
the same point:
where ∇ is the gradient operator. This is
equivalent to the force definition above,
since electric potential Φ is defined by
the electric potential energy U per unit positive
charge:
and force is the negative of potential energy
gradient:
If several spatially distributed charges generate
such an electric potential, e.g. in a solid,
an electric field gradient may also be defined.
Uniform fields
A uniform field is one in which the electric
field is constant at every point. It can be
approximated by placing two conducting plates
parallel to each other and maintaining a voltage
between them; it is only an approximation
because of edge effects. Ignoring such effects,
the equation for the magnitude of the electric
field E is:
where Δϕ is the potential difference between
the plates and d is the distance separating
the plates. The negative sign arises as positive
charges repel, so a positive charge will experience
a force away from the positively charged plate,
in the opposite direction to that in which
the voltage increases. In micro- and nanoapplications,
for instance in relation to semiconductors,
a typical magnitude of an electric field is
in the order of 1 volt/µm achieved by applying
a voltage of the order of 1 volt between conductors
spaced 1 µm apart.
Parallels between electrostatic and gravitational
fields
Coulomb's law, which describes the interaction
of electric charges:
is similar to Newton's law of universal gravitation:
.
This suggests similarities between the electric
field E and the gravitational field g, so
sometimes mass is called "gravitational charge".
Similarities between electrostatic and gravitational
forces:
Both act in a vacuum.
Both are central and conservative.
Both obey an inverse-square law.
Differences between electrostatic and gravitational
forces:
Electrostatic forces are much greater than
gravitational forces for natural values of
charge and mass. For instance, the ratio of
the electrostatic force to the gravitational
force between two electrons is about 1042.
Gravitational forces are attractive for like
charges, whereas electrostatic forces are
repulsive for like charges.
There are not negative gravitational charges
while there are both positive and negative
electric charges. This difference, combined
with the previous two, implies that gravitational
forces are always attractive, while electrostatic
forces may be either attractive or repulsive.
Electrodynamic fields
Electrodynamic fields are E-fields which do
change with time, when charges are in motion.
An electric field can be produced not only
by a static charge, but also by a changing
magnetic field. The electric field is then
given by:
in which B satisfies
and ∇× denotes the curl. The vector field
B is the magnetic flux density and the vector
A is the magnetic vector potential. Taking
the curl of the electric field equation we
obtain,
which is Faraday's law of induction, another
one of Maxwell's equations.
Energy in the electric field
The electrostatic field stores energy. The
energy density u is given by
where ε is the permittivity of the medium
in which the field exists, and E is the electric
field vector.
The total energy U stored in the electric
field in a given volume V is therefore
Further extensions
Definitive equation of vector fields
In the presence of matter, it is helpful in
electromagnetism to extend the notion of the
electric field into three vector fields, rather
than just one:
where P is the electric polarization – the
volume density of electric dipole moments,
and D is the electric displacement field.
Since E and P are defined separately, this
equation can be used to define D. The physical
interpretation of D is not as clear as E or
P, but still serves as a convenient mathematical
simplification, since Maxwell's equations
can be simplified in terms of free charges
and currents.
Constitutive relation
The E and D fields are related by the permittivity
of the material, ε.
For linear, homogeneous, isotropic materials
E and D are proportional and constant throughout
the region, there is no position dependence:
For inhomogeneous materials, there is a position
dependence throughout the material:
For anisotropic materials the E and D fields
are not parallel, and so E and D are related
by the permittivity tensor, in component form:
For non-linear media, E and D are not proportional.
Materials can have varying extents of linearity,
homogeneity and isotropy.
See also
Classical electromagnetism
Magnetism
Teltron Tube
Teledeltos, a conductive paper that may be
used as a simple analog computer for modelling
fields.
References
External links
Electric field in "Electricity and Magnetism",
R Nave – Hyperphysics, Georgia State University
'Gauss's Law' – Chapter 24 of Frank Wolfs's
lectures at University of Rochester
'The Electric Field' – Chapter 23 of Frank
Wolfs's lectures at University of Rochester
[1] – An applet that shows the electric
field of a moving point charge.
Fields – a chapter from an online textbook
Learning by Simulations Interactive simulation
of an electric field of up to four point charges
Java simulations of electrostatics in 2-D
and 3-D
Electric Fields Applet – An applet that
shows electric field lines as well as potential
gradients.
Interactive Flash simulation picturing the
electric field of user-defined or preselected
sets of point charges by field vectors, field
lines, or equipotential lines. Author: David
Chappell
