Maxwell's Equations
Maxwell's equations are the
set of four equations by James Maxwell that describe the behavior of
both the electric and magnetic fields. Maxwell's equations provided the
basis for the unification of electric field and magnetic field, the
electromagnetic description of light, and ultimately, Albert Einstein's
theory of relativity.
The elegant modern mathematical formulations of Maxwell's equations
were not developed by Maxwell. In 1884, Oliver Heaviside reformulated
Maxwell's equations using vector calculus. This change reinforced the
perception of physical symmetries between the various fields with a more
symmetric mathematical representation.
The electromagnetic field equations have an intimate link with special
relativity: the magnetic field equations can be derived from consideration
of the transformation of the electric field equations under relativistic
transformations at low velocities.
 ·D
= ρ
where ρ is the electric charge density (in units of C/m3),
and D is the electric displacement field (in units of C/m2)
which is related to the electric field E via a materials-dependent
constant called the permittivity, ε. The permittivity of free space is
referred to as ε0, resulting in the equation for
free space:
·E
= ρ/ε0
where, again, E is the electric field (in units of V/m), ρ is
the charge density, and ε0 (approximately 8.854
pF/m) is the permittivity of free space.
Equivalent integral form: ∫AE·dA = Qenclosed
/ ε0
dA is the area of a differential square on the surface A with an
outward facing surface normal defining its direction, Qenclosed
is the charge enclosed by the surface.
Note: the integral form only works if the integral is over a closed
surface. Shape and size do not matter. The integral form is also known as
Gauss's Law.
This equation corresponds to Coulomb's law.
·B
= 0
B is the magnetic flux (in units of tesla, T).
Equivalent integral form: ∫AB·dA = 0
dA is the area of a differential square on the surface A with an
outward facing surface normal defining its direction.
Note: like the electric field's integral form, this equation only works
if the integral is done over a closed surface.
This equation is related to the magnetic field's structure because it
states that given any volume element, the net magnitude of the vector
components that point outward from the surface must be equal to the net
magnitude of the vector components that point inward. Structurally, this
means that the magnetic field lines must be closed loops. Another way of
putting it is that the field lines cannot originate from somewhere;
attempting to follow the lines backwards to their source or forward to
their terminus ultimately leads back to the starting position. This
implies that there are no magnetic monopoles. If a monopole were to be
discovered, this equation would need to be modified to read
·B
= ρm
where ρm would be the density of magnetic monopoles.
×E
= -∂B/∂t
Equivalent Integral Form: ε = -dφB/dt where φB
= ∫AB·dA
φB is the magnetic flux through the area A described
by the second equation, ε is the Electromotive Force around the edge of
the surface A.
Note: this equation only works of the surface A is not closed
because the net magnetic flux through a closed surface will always be
zero, as stated by the previous equation. That, and the electromotive
force is measured along the edge of the surface; a closed surface has no
edge. Some textbooks list the Integral form with an N (representing the
number of coils of wire that are around the edge of A) in front of the
flux derivative. The N can be taken care of in calculating A (multiple
wire coils means multiple surfaces for the flux to go through), and it is
an engineering detail so it has been omitted here.
Note the negative sign; it is necessary to maintain conservation of
energy. It is so important that it even has its own name, Lenz's Law.
This equation relates the electric and magnetic fields, but it also has
a lot of practical applications, too. This equation describes how electric
motors and electric generators work.
This law corresponds to the Faraday's law of electromagnetic induction.
×H
= J + ∂E/∂t
where H is the magnetic field strength (in units of A/m),
related to the magnetic flux B by a constant called the
permeability, μ, and J is the
current density,
defined by: J = ∫ρqvdV where v is a vector
field called the drift velocity that describes the velocities of that
charge carriers which have a density described by the scalar function ρq.
In free space, the permeability μ is the permeability of free space, μ0,
which is defined to be exactly 4π×10-7 W/Am. Thus, in
free space, the equation becomes:
×B
= μ0J + μ0ε0∂E/∂t
Equivalent integral form: ∫sB·ds = μ0Iencircled
- μ0ε0∫A (∂E/∂t)·dA
s is the edge of the open surface A (any surface with the curve s as
its edge will do), and Iencircled is the current encircled by
the curve s (the current through any surface is defined by the equation: Ithrough
A = ∫AJ·dA).
Note: unless there is a capacitor or some other place where
·J
≠ 0, the second term on the right hand side is generally negligible and
ignored. Any time this applies, the integral form is known as Ampere's
Law.
Summary
-
·D
= ρ
-
·B
= 0
-
×E
= -∂B/∂t
-
×H
= J + ∂D/∂t
For free space, eliminating the nonphysical quantities D and
H, this reduces to:
-
·E
= ρ/ε0
-
·B
= 0
-
×E
= -∂B/∂t
-
×B
= μ0J + μ0ε0∂E/∂t
Simplifying further, by considering the case in the absence of imposed
current or electric charge, gives the propagation equation for
electromagnetic waves in free space:
-
·E
= 0
-
·B
= 0
-
×E
= -∂B/∂t
-
×B
= μ0ε0∂E/∂t
This equation has a simple solution in terms of traveling sinusoidal
plane waves, traveling at the speed (μ0ε0)-1/2.
Maxwell's observation that μ0ε0
= c-2, relating the speed of light c to the permittivity and
permeability of free space, was the first confirmation that light was
electromagnetic radiation.
A Final Note on Unit Systems
The above equations are all in a unit system called mks (short for
meter, kilogram, second; also know as the International System of Units
(or SI for short). This is more commonly known as the metric system. In a
related unit system, called cgs (short for centimeter, gram, second), the
equations take on a more symmetrical form, as follows:
-
·E
= 4πρ
-
·B
= 0
-
×E
= -c-1 ∂B/∂t
-
×B
= c-1 ∂E/∂t + 4πc-1J
Where c is the speed of light in a vacuum. The symmetry is more
apparent when the electromagnetic field is considered in a vacuum. The
equations take on the following, highly symmetric form:
Note: All variables that are in bold represent vector
quantities; see also vector calculus.
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.
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