Lets recall Maxwell's equations in differential form
In
free space there are no charges or currents these become:
Lets take the time derivative of
but recall
so using that and
we get
This is the 3d wave equation! Note that is a second time derivative on
one side and a second space derivative on the other side
It is left as an exercise to show that
we also see from this equation that the speed of light in vacuum is
We want to find the expression for a plane that is perpendicular to
,
where
is a vector in the direction of propagation of the wave.The plane is the
set of points that has the same projection onto the vector
That is any point
that satisfies
is a point on the planeNow consider the function
we see that the magnitude of
is the same over every plane that is defined by
we want to construct harmonic waves, ie. they should repeat every
wavelength along the direction of propagation so they should satisfy
where
λ
is the wavelengththen we must have
This is true if
eiλk
=
1
=
ei
2
π
or
λk
=
2
π
This should have a familiar look to it! Finally we want these waves to
propagate in time so you should be able to guess the answer from our work on
mechanical waves
To find spherical solutions to the wave equation it is natural to use spherical coordinates. x = r sinθ cosφ y = r sinθ sinφ z = r cosθ
There is a very nice discussion of Spherical Coordinates at:
http://mathworld.wolfram.com/SphericalCoordinates.html
There is also a nice discussion of Cylindrical coordinates at the same site
http://mathworld.wolfram.com/CylindricalCoordinates.html
Beware the confusion about θ and φ . We are calling the polar angle θ . All other mathematical disciplines get it wrong and call it φ .
The Laplacian can be written in spherical coordinates, but where does that
come from?looking at just the
x
term
Then you take the second derivative to get
which as you can imagine is a tremendously boring and tedious thing to
do.Since this isn't a vector calculus course lets just accept the
solution.In the case of spherical waves it is not so difficult
since the
θ
and
φ
derivative terms all go to
0
.
Thus for spherical waves, we can write the wave equation:
Now we can multiply both sides by
r
and since
r
does not depend upon
t
write
This is just the one dimensional wave equation with a harmonic solution
r
ψ
(
r
,
t
)
=
A
ei
k
(
r
∓
v
t
)
or
A plane wave solution to the electromagnetic wave equation for the
field is
In
vacuum with no currents present we know that:
.
Recall that earlier we showed
So
implies
that the
associated with our plane wave is perpendicular to its direction of motion.
Likewise
implies that the
field is also perpendicular to the direction of motion Lets pick a
specific simple case:
Then
Faraday's law
tells
us that (since
)
That
is the
field is at Right angles to the
field.Also
I
leave as an exercise showing
A movie demonstrating a plane wave can be seen at
http://www.cs.brown.edu/stc/outrea/greenhouse/nursery/physics/gfx/emwave.mov
An applet can be viewed at
The electric and magnetic fields have energy and hence have an energy density.
We can see this for a capacitor:The energy stored in a capacitor is
where
C
is the capacitance and
V
the potential drop (voltage) across the capacitor. For a parallel plate
capacitor
and
V
=
Ed
where
A
is the area of the plates
d
the distance between them and
E
the electric field strength.note that
Ad
is the volumeThus
So we can write the energy density (Energy per Unit volume) of the field as
Likewise by calculating the energy stored by a B-field in a current
carrying solenoid one can derive:
Since we know
E
=
cB
In an EM wave
u
=
uE
+
uB
which is
u
=
ε0E2
or equivalently
u
=
B2
/
μ0
Lets take some time to review what we have learned so far. We have derived
Maxwell's equations in differential form.
These, in general are much more useful than the integral form you learned in
Freshman Physics. These allow one to understand the relationship between
fields, charges and currents as a function of position. This point by point
understanding of what is happening is not obvious in the integral form of the
equations.
Another interesting point is that if everything is static, that is nothing is
changing with time, then they
become
Notice that for static fields, there is no interplay between electricity and
magnetism. If there was just electrostatics, then we would have separate
electric and magnetic fields. Maxwell was able to show that the electricity
and magnetism are intimately related, and the theory is unified in that you
need both. (To this day the unification of forces is one of the driving
principles of a lot of physics research - I would say the only interesting
physics research but that is perhaps because I do it for a living.)
In free space Maxwell's equations become:
We then showed that one can take time derivatives and end up with
which is the 3d wave equation! Note that is a second time derivative on
one side and a second space derivative on the other side, the hallmarks of a
wave equation.
It was left as an exercise to show that
We also see from this equation that the speed of light in vacuum is
A plane wave solution to the electromagnetic wave equation for the
field is
In vacuum with no currents present we know that:
.
Recall that earlier we showed
So
implies that the
associated with our plane wave is perpendicular to its direction of motion.
Likewise
implies that the
field is also perpendicular to the direction of motion
The electric and magnetic fields have energy and hence have an energy density. In an EM wave u = uE + uB which is u = ε0E2 or equivalently u = B2 / μ0 . This is all very amazing when you think about it. Maxwell's equations tell us that we can have waves in the electric and magnetic fields. These waves carry energy. That is they are a mechanism to transport energy through free space (or a medium). This is why the sun warms us, which is pretty important.
Now we want to calculate the power crossing a given area
A
.
During a time
Δ
t
an EM wave will pass an amount of energy through
A
of
u
c
Δ
t
A
where
u
is the energy density of the wave. If we want the
power/
m2
then we must divide by
Δ
t
A
.
Thus we get
Now we make the reasonable assumption that the energy flows in the direction
of the wave, ie. perpendicular to
and