Notes on Relativity and Cosmology - Physics Department, UCSB

2.1. THE BASICS OF E & M 47
(moving charges) when that current is in a vacuum 3 . The key point here is that
both of the numbers are things that had been measured in the laboratory long
before Maxwell or anybody else had ever thought of ‘electromagnetic waves.’
−12 C2
−7 Ns2
Their values were ǫ 0 = 8.854 × 10
Nm
and µ 2 0 = 4π × 10
C
. 2
Anyway, this other Equation of Maxwell’s looks like:
∂E
∂t = ǫ 0µ 0
∂B
∂x . (2.2)
Now, to understand how the waves come out of all this, it is useful to take the
derivative (on both sides) of equation (2.1) with respect to time. This yields
some second derivatives:
∂ 2 B
∂t 2 = ∂2 E
∂x∂t
(2.3)
Note that on the right hand side we have taken one derivative with respect to
t and one derivative with respect to x.
Similarly, we can take a derivative of equation (2.2) on both sides with respect
to x and get:
∂ 2 B
∂x 2 = (ǫ 0µ 0 ) ∂2 E
∂x∂t . (2.4)
Here, I have used the interesting fact that it does not matter whether we first
∂ ∂
differentiate with respect to x or with respect to t:
∂t ∂x E = ∂ ∂
∂x ∂t E.
Note that the right hand sides of equations (2.3) and (2.4) differ only by a factor
of ǫ 0 µ 0 . So, I could divide equation (2.4) by this factor and then subtract it
from (2.3) to get
∂ 2 B
∂t 2 − 1 ∂ 2 B
ǫ 0 µ 0 ∂x 2 = 0 (2.5)
This is the standard form for a so-called ‘wave equation.’ To understand why,
let’s see what happens if we assume that the magnetic field takes the form
B = B 0 sin(x − vt) (2.6)
for some speed v. Note that equation (2.6) has the shape of a sine wave at any
time t. However, this sine wave moves as time passes. For example, at t = 0
the wave vanishes at x = 0. On the other hand, at time t = π/2v, at x = 0 we
3 Charges and currents placed in water, iron, plastic, and other materials are associated
with somewhat different values of electric and magnetic fields, described by parameters ǫ and
µ that depend on the materials. This is due to what are called ‘polarization effects’ within the
material, where the presence of the charge (say, in water) distorts the equilibrium between
the positive and negative charges that are already present in the water molecules. This is
a fascinating topic (leading to levitating frogs and such) but is too much of a digression to
discuss in detail here. See PHY212 or the advanced E & M course. The subscript 0 on ǫ 0 and
µ 0 indicates that they are the vacuum values or, as physicists of the time put it, the values
for ‘free space.’