Notes on Relativity and Cosmology - Physics Department, UCSB

2.1. THE BASICS OF E & M 45
⋆ Conversely, Maxwell discovered that an electric field which changes in time
produces a magnetic field. Maxwell codified both this observation and Faraday’s
law in a set of equations known as, well, Maxwell’s equations. Thus, a field that
is purely electric in one reference frame will have a magnetic part in another
frame of reference.
As a result, it is best not to think of electricity and magnetism as separate phenomena.
Instead, we should think of them as forming a single “electromagnetic”
field which is independent of the reference frame. It is the process of breaking
this field into electric and magnetic parts which depends on the reference frame.
There is a strong analogy 2 with the following example: The spatial relationship
between the physics building and the Hall of Languages is fixed and independent
of any coordinate system. However, if I am standing at the Physics building
and want to tell someone how to walk to HOL, depending on which direction I
am facing I may tell that person to “walk straight ahead across the quad,” or I
might tell them to “walk mostly in the direction I am facing but bear a little to
the right.” The relationship is fixed, but the description differs. For the moment
this is just a taste of an idea, but we will be talking much more about this in
the weeks to come. In the case of electromagnetism, note that this is consistent
with the discovery that magnetic charge is really moving electric charge.
Not only do we find a conceptual unity between electricity and magnetism, but
we also find a dynamical loop. If we make the electric field change with time
in the right way, it produces a magnetic field which changes with time. This
magnetic field then produces an electric field which changes with time, which
produces a magnetic field which changes with time..... and so on. Moreover, it
turns out that a changing field (electric or magnetic) produces a field (magnetic
or electric) not just where it started, but also in the neighboring regions of space.
This means that the disturbance spreads out as time passes! This phenomenon
is called an electromagnetic wave. Electromagnetic waves are described in more
detail below in section 2.1.1, and all of their properties follow from Maxwell’s
equations. For the moment, we merely state an important property of electromagnetic
waves: they travel with a precise (finite) speed. See section 2.1.1 for
the derivation.
2.1.1 Maxwell’s Equations and Electromagnetic Waves
The purpose of this section is to show you how Maxwell’s equations lead to
electromagnetic waves (and just what this means). This section is more mathematical
than anything else we have done so far, and I will probably not go
through the details in class. If you’re not a math person, you should not be
scared off by the calculations below. The important point here is just to get
the general picture of how Maxwell’s equations determine that electromagnetic
waves travel at a constant speed.
2 This analogy may be clear as mud at the moment, but will be clearer later in the course
as we think about a number of similar effects.