174 Maxwell's Equations
scopic, or averaged-out, version V = IR, where V is the voltage measured
in volts, / is the current in amperes, and R is the resistance in ohms.
If an insulated conductor is placed in an external electric field, then the
induced electric current redistributes the charged particles on the surface
of the conductor so that, in equilibrium, the total electric field inside the
conductor is zero: if there were non-zero field inside the conductor, then,
by Ohm's Law, there would be a non-zero current, which is impossible
for an isolated conductor in equilibrium. The force lines of the electric
field are therefore normal to the surface of the conductor; otherwise, the
tangential component would produce a surface current. There is no charge
accumulations inside the conductor either, as such accumulations would
produce electric current. Thus, a closed conducting surface isolates the
interior from the exterior electric field. For that reason, delicate micro-
chips and electronic equipment are often stored and shipped in aluminum
foil or other conducting cover.
In dielectrics, such as glass, plastics, and rubber, there is little or no
free motion of charged particles when an external electric field is applied.
Instead, the external field penetrates the material, causing polarization of
molecules in the substance and producing numerous point electric dipoles.
In biological materials, cell membranes are good insulators against ion flux,
and ions form dipoles near the surface of cell membranes. Protein molecules
form dipoles by ionization of the amino-acid chains.
Denote by P the resulting density of dipole moments per unit volume
in the dielectric material. Recall that a point dipole has zero net charge;
direct computations show that the flux of the corresponding electric field E
(see (3.3.29) through every closed piece-wise smooth surface is zero, as long
as the dipole is not on the surface. Accordingly, instead of looking at the
flux of E, we will look at the flux of P. Note that P has the dimension of
charge per unit area, and so the divergence div P of P has the dimension
of charge per unit volume. Recall that the direction of the dipole moment
is from the negative charge to the positive. It is then natural to define the
scalar field pb so that pb = — div P. With this definition, if G is any part
of the dielectric, then ///(divP — pb) dV = 0, which is consistent with
G
the idea that there is no movement of charges inside the dielectric. The
scalar field pb is called the density of bound charges. It is also possible
that some free charges are present inside the dielectric, with density per
unit volume pf. By Gauss's Law of Electric Flux (3.3.9), a region in free
space with density pb + p/ of charges per unit volume produces the electric