a/A capacitor with a dielec-
tric between the plates.only penetrates to a very small depth, called the skin depth. In
the limit of poor conduction and low frequencies, absorption pre-
dominates, and the skin depth becomes much greater. In a high-
frequency AC circuit, the skin depth in a copper wire is very small,
and therefore the signals in such a circuit are propagated entirely at
the surfaces of the wires. In the limit of low frequencies, i.e., DC,
the skin depth approaches infinity, so currents are carried uniformly
over the wires’ cross-sections.
We can quantify how well a particular material conducts elec-
tricity. We know that the resistance of a wire is proportional to its
length, and inversely proportional to its cross-sectional area. The
constant of proportionality is 1/σ, whereσ(not the sameσas the
surface charge density) is called the electrical conductivity. Exposed
to an electric fieldE, a conductor responds with a current per unit
cross-sectional areaJ = σE. The skin depth is proportional to
1 /√
fσ, wherefis the frequency of the wave.11.7.2 Dielectrics
A material with a very low conductivity is an insulator. Such
materials are usually composed of atoms or molecules whose elec-
trons are strongly bound to them; since the atoms or molecules
have zero total charge, their motion cannot create an electric cur-
rent. But even though they have zero charge, they may not have
zero dipole moment. Imagine such a substance filling in the space
between the plates of a capacitor, as in figure a. For simplicity,
we assume that the molecules are oriented randomly at first, a/1,
and then become completely aligned when a field is applied, a/2.
The effect has been to take all of the negatively charged black ands
of the molecules and shift them upward, and the opposite for the
positively charged white ends. Where the black and white charges
overlap, there is still zero net charge, but we have a strip of nega-
tive charge at the top, and a strip of positive charge at the bottom,
a/3. The effect has been to cancel out part of the charge that was
deposited on the plates of the capacitor. Now this is very subtle, be-
cause Maxwell’s equations treat these charges on an equal basis, but
in terms of practical measurements, they are completely different.
The charge on the plates can be measured be inserting an ammeter
in the circuit, and integrating the current over time. But the charges
in the layers at the top and bottom of the dielectric never flowed
through any wires, and cannot be detected by an ammeter. In other
words, the total charge,q, appearing in Maxwell’s equartions is ac-
tualyq=qfree−qbound, whereqfreeis the charge that moves freely
through wires, and can be detected in an ammeter, whileqboundis
the charge bound onto the individual molecules, which can’t. We
will, however, detect the presence of the bound charges via their
electric fields. Since their electric fields partially cancel the fields of
the free charges, a voltmeter will register a smaller than expected
Section 11.7 Electromagnetic properties of materials 735