Notes on Relativity and Cosmology - Physics Department, UCSB

46 CHAPTER 2. MAXWELL, E&M, AND THE ETHER
Below, I will not use the ‘complete’ set of Maxwell’s equations – instead, I’ll
use a slightly simplified form which is not completely general, but which is
appropriate to the simplest electro-magnetic wave. Basically, I have removed
all of the complications having to do with vectors. You can learn about such
features in PHY212 or, even better, in the upper division electromagnetism
course.
Recall that one of Maxwell’s equations (Faraday’s Law) says that a magnetic
field (B) that changes is time produces an electric field (E). I’d like to discuss
some of the mathematical form of this equation. To do so, we have to turn
the ideas of the electric and magnetic fields into some kind of mathematical
objects. Let’s suppose that we are interested in a wave that travels in, say, the
x direction. Then we will be interested in the values of the electric and magnetic
fields at different locations (different values of x) and a different times t. As a
result, we will want to describe the electric field as a function of two variables
E(x, t) and similarly for the magnetic field B(x, t). (As mentioned above, we
are ignoring the fact that electric and magnetic fields are vector quantities; that
is, that they are like arrows that point in some direction. For the vector experts,
I have just picked out the relevant components for discussion here.)
Now, Faraday’s law refers to magnetic fields that change with time. How fast
a magnetic field changes with time is described by the derivative of the magnetic
field with respect to time. For those of you who have not worked with
‘multivariable calculus,’ taking a derivative of a function of two variables like
B(x, t) is no harder than taking a derivative of a function of one variable like
y(t). To take a derivative of B(x, t) with respect to t, all you have to do is to
momentarily forget that x is a variable and treat it like a constant. For example,
suppose B(x, t) = x 2 t + xt 2 . Then the derivative with respect to t would be
just x 2 + 2xt. When B is a function of two variables, the derivative of B with
respect to t is written ∂B
∂t .
It turns out that Faraday’s law does not relate ∂B
∂t
directly to the electric field.
Instead, it relates this quantity to the derivative of the electric field with respect
to x. That is, it relates the time rate of change of the magnetic field to the way
in which the electric field varies from one position to another. In symbols,
∂B
∂t = ∂E .. (2.1)
∂x
It turns out that another of Maxwell’s equations has a similar form, which
relates the time rate of change of the electric field to the way that the magnetic
field changes across space. Figuring this out was Maxwell’s main contribution
to science. This other equation has pretty much the same form as the one above,
but it contains two ‘constants of nature’ – numbers that had been measured in
various experiments. They are called ǫ 0 and µ 0 (‘epsilon zero and mu zero’).
The first one, ǫ 0 is related to the amount of electric field produced by a charge
of a given strength when that charge is in a vacuum. Similarly, µ 0 is related to
the amount of magnetic field produced by a certain amount of electric current