Let us first consider vacuum.
Light is an electromagnetic wave, propagating in space and time at a constant
speed c, in such a way that spatial variations along the direction of
propagation in the electric field generate time variations of the magnetic
field, and changes in the electric field along the direction of time cause
spatial changes of the magnetic field along the direction of propagation. You
could then say that the electric and magnetic components of light are
interrelated through the space-time continuum. This is it! We have just
stated two of Maxwell's equations. Now how about the other two? Well, they just
say that there are no sources or sinks of the magnetic and electric fields in
vacuum. In other words, that the fields don't just appear or disappear in empty
space. Which is hardly surprising.
In their mathematical form Maxwell's equation are written as:
(Eq.7.1a)
(Eq.7.1b)
(Eq.7.1c)
(Eq.7.1d)
I have chosen the CGS units instead of the SI because they
are more symmetric and perhaps easier to grasp intuitively.
In order to understand these formula, you should of course
understand the meaning of the divergence and curl operators.
7.1 Divergence
The divergence of a field
F is a measure of sources and sinks for that field. For instance the light
intensity generated by a candle or an electrical bulb has a large divergence -
if you look at its value along any axis in space you will measure a decrease
with distance. An inflating balloon also has a large divergence - the amount of
its material along any Cartesian axis gets thinner and thinner as the balloon
expands. This is expressed mathematically as:
(Eq.7.2)
Which is the vector product of the vector gradient
defined as:
(Eq.7.3)
with the field vector F.
7.2 Curl
The curl of a field F is a vector. It is a measure of
the rotation the field has at any point. The direction of the curl is determined
according to the right hand rule. The magnitude of the curl tells us how much
rotation there is.
In order to visualize the curl of a field, the field is often
represented as a water flow. One can then place an imaginary paddle wheel at
different places in the flow and estimate the rate of rotation of the paddle
wheel (see Fig. 7).
Fig.
7 Illustration of the curl of the electric field in an electromagnetic wave.
The electric field is represented in blue and the magnetic one is in red.
Is there an intuitive way to relate the curl of the electric
field with the magnetic field in a way to produce equation 7.1a? Well, there are
several observations that we can make after looking at Fig.7. First, we notice
that the vectors generated by the curl of E are in the plane of B.
So far so good!
Second, we see that they have an opposite sign to B.
So, we can introduce a minus sign on the left or the right side of our equation
and write:
Third, the vectors generated by the curl of E appear
to be phase shifted by 90° with respect to B. Now, we know that B
has a sinusoidal form, and a wave that has a 90° phase shift with respect to a
sine curve is, of course, a cosine curve. And how do we go from a sine to a
cosine curve (or vice-versa)? We use the derivative! Hence:
Finally, the vectors generated by the curl of E have a
different amplitude from B. Hum, lets see... on the one side we have
variations in space, on the other we have variations in time. All we have to do
is to equilibrate the units of this equation. And the most natural way to do
that is by introducing the constant c - the speed of light. And voila:
(Eq.7.1a)
Eq. 7.1b can be obtained in a similar way.
Mathematically, the curl is defined as the cross product of
the vector gradient with the field vector F:
(Eq.7.4)
In a sense,
curl and div are complementary. The latter requires that the
field changes when moving along the field direction, the former that the field
varies when moving across the field direction. This is also understandable since
the divergence is the vector product with the gradient while the
curl
is the cross product with the gradient.
