- Maxwell's Equations
- Differential Form of Maxwell's Equations
- Electromagnetic Wave Equation
- Light as Transverse Waves
- Orthogonality of the Electric and Magnetic Fields
- Energy in an Electromagnetic Wave

, (4.1)

where *q* is the electrical charge on the
object in question and *E* is the **electric field** produced by
all the other charges in the universe. The charge was discovered to take
on a discrete set of values, one of the first examples of quantization.
In its turn, the electric field can be described by a scalar potential
field *V*, which is related to the electric field by

. (4.2)

In addition, it was also noted that a moving
charge may experience another force which is proportional to its velocity
** v**. This led to the definition of another field; namely the
magnetic field

. (4.3)

As with the electric field, the magnetic field
is generated by all the other currents in the universe. The magnetic field
can be described in terms of a vector potential field *A*, which is
related to the magnetic field by

. (4.4)

If the electric and magnetic forces occur concurrently,
then the force on the charge is given by the **Lorentz force law**

. (4.5)

, (4.6)

, (4.7)

, (4.8)

. (4.9)

Each one of these can be understood separately.

The first of Maxwell's equations, equation (4.6),
is known as **Gauss's Law**. It relates the flux of electric field intensity
to the total charge enclosed by the surface. The flux is defined as

, (4.10)

where *d S* is the vector outwardly
normal to the surface and the integral is over the entire surface enclosing
the region in question. In words, Gauss's law tells us that the total flux
through a closed surface, i.e. the change in the number of field lines
passing through a closed surface, is proportional to the total charge contained
within the volume defined by the surface. Thus if there is no charge inside
the surface, the net flux is zero. If there is a positive net charge, the
enclosed region acts as a

The constant
is called the **electric permittivity** of the medium. If the medium
is a vacuum, then
= _{0},
where _{0}
is known as the **permittivity of free space** and has a value of .
The electric permittivity was originally used to act as a medium dependent
proportionality constant that connects a parallel plate capacitor's capacitance
with its geometric characteristics. Conceptually, we can view the permittivity
as encompassing the electrical behavior of the medium: in a sense, it is
a measure of the degree to which the material is permeated by the electric
field in which it is immersed. We can relate the electric permittivity
to the dielectric constant by the following formula

. (4.11)

The second equation is also a form of Gauss's law, this time applied to the magnetic field. The fact that the enclosed charge is zero tells us that, at least according to classical electromagnetic theory, there is no such thing as a magnetic monopole. In other words, whereas the electrical charge could be viewed as either a positive or negative charge individually, we can never find magnetic charges which do not include both a positive and negative pole. Since the total enclosed charge is the algebraic sum of the charges, this lack of magnetic monopoles automatically insures that the sum is zero.

The third equation is known as **Faraday's law**.
In a manner similar to the electric flux, the magnetic flux is defined
as

, (4.12)

where the surface is now an open surface bounded by a conducting loop. Faraday found that if the induced emf that was developed in the loop depended on the rate at which the magnetic flux changed,

. (4.13)

However, the emf exists only as a result of the presence of an electric field, which is related to the emf by

. (4.14)

Combining (4.13) and (4.14), any direct reference to the induced emf is removed and we get Faraday's law. Physically, this shows us that if the magnetic flux changes, in other words if either the surface area or the magnetic field changes with time, then an electrical field is produced as result. This electrical field creates an emf which acts in such a way as to resist the changes in the magnetic flux. Thus, a time varying magnetic field creates an electric field. Since there are no charges which act as a source or a sink, the field lines close on themselves, forming loops.

The last of Maxwell's equations is known as **Ampere's
Law**. In its original form as expressed by Ampere, it related the number
of magnetic field lines which passed through a surface formed by a closed
loop to the total amount of current which was enclosed by the loop

, (4.15)

where ** j** is known as the current
density. The open surface is bounded by the loop, and the quantity
is called the

, (4.16)

where *K _{B}* is called the relative
permeability. In a manner similar to the dielectric constant, the relative
permeability can be viewed as a measurement of how well the magnetic field
permeates a material.

While Ampere's law in its original formulation
explained many important effects, such as the operation of a solenoid,
it was found to also create larger problems. In particular, use of Ampere's
law in the form of equation (4.15) led to violation of conservation of
energy for the electric and magnetic fields. In order to correct this,
Maxwell hypothesized the existence of an additional current, the **displacement
current**, which is defined as

. (4.17)

When this is combined with Ampere's law in a region with no physical currents, we get

In other words, just as a time varying magnetic flux lead to the creation of a circular electric field, so to does a time varying electric flux lead to the creation of a linear magnetic field. If a physical current also exists, we again regain the last of Maxwell's equations.

. (4.18)

Similarly, Stokes theorem states that the flux through a closed loop is equal the integral of the curl of the field over the area enclosed by the loop

. (4.19)

Let's start with Gauss's divergence theorem and apply it to the first two of Maxwell's equations. Then we get

and

These relations must be equal for any volume, so the first two Maxwell's equations become

(4.20)

and

. (4.21)

Applying Stokes theorem to the last two of Maxwell's equations yields

and

These relations must hold for any surface bounded by a closed loop, so the last two Maxwell's equations become

(4.22)

and

. (4.23)

To derive the wave equation for the electric field, start with the third of Maxwell's equations and take the curl of both sides

. (4.24)

The left hand side can be simplified by using the vector relationship

(4.25)

to get

, (4.26)

where the last step used the fact that . To evaluate the right hand side of (4.24), we start with the fact that the spatial derivatives () and the time derivative can be interchanged. We then use the last of Maxwell's equations to find

. (4.27)

Combining (4.26) and (4.27), we get

, (4.28)

which we recognize as the three dimensional wave equation for each component of the electric field. Comparing (4.28) with the standard result for a wave, we see that

. (4.29)

Using the fact that the experimentally determined
speed of light is also 3.00 x 10^{8} m/s, we are lead to the inescapable
conclusion that light is just one form of electromagnetic wave propagation.
When the electromagnetic disturbance is moving in a vacuum, we denote its
speed by a special symbol, *c*.

In a manner similar to those leading to (4.28), we can start with the last of Maxwell's equations to find the wave equation for the magnetic field. Thus,

. (4.30)

From this, we see that the electromagnetic wave has no electric field component in the direction of propagation. Thus, the electric field is exclusively transverse. A similar argument can be used on Gauss's law for magnetic fields to show that it is also transverse to the direction of propagation. In particular, Faraday's law tells us that

. (4.31)

In other words, the time dependent magnetic field
can only have a component in the *z* direction when the electric field
is exclusively in the *y* direction. From these, we see that, **in
free space, the plane electromagnetic wave is transverse**.

. (4.32)

Then, from (4.31), we have that

. (4.33)

Comparing (4.32) and (4.33), we see that

. (4.34)

Notice that even though this was derived using
a plane wave, (4.34) is true for waves moving in the *x* direction
in general.

. (4.35)

Similarly, the energy density stored in the magnetic field is

. (4.36)

Since , we have that

. (4.37)

Thus, the total energy density is shared between the constituent electric and magnetic fields

. (4.38)

To represent the flow of electromagnetic energy,
let *S* symbolize the transport of energy per unit area. During a
very small interval of time *t*,
only the energy contained in the volume *V* = *cAt*
will cross the area *A*. Thus,

. (4.39)

We now assume that, for an isotropic media, the
energy flows in the direction of propagation of the wave. Then the corresponding
vector ** S** is

. (4.40)

This vector is known as the **Poynting vector**.
If ** E** and

, (4.41)

so

. (4.42)

Since the electric field is considerably more
effective at exerting forces and doing work on charges than the magnetic
field, the electric field ** E** is referred to as the

Last updated: June 28, 1997

Comments to: D-Suson@tamuk.edu