i/Applying Gauss’ law to an
infinite line of charge.
with charge, change the Coulomb constantkto the gravitational
constantG, and insert a minus sign because the gravitational fields
around a (positive) mass point inward.
Gauss’ theorem can only be proved if we assume a 1/r^2 field,
and the converse is also true: any field that satisfies Gauss’ theo-
rem must be a 1/r^2 field. Thus although we previously thought of
Coulomb’s law as the fundamental law of nature describing electric
forces, it is equally valid to think of Gauss’ theorem as the basic law
of nature for electricity. From this point of view, Gauss’ theorem is
not a mathematical fact but an experimentally testable statement
about nature, so we’ll refer to it as Gauss’law, just as we speak of
Coulomb’slawor Newton’slawof gravity.
If Gauss’ law is equivalent to Coulomb’s law, why not just use
Coulomb’s law? First, there are some cases where calculating a
field is easy with Gauss’ law, and hard with Coulomb’s law. More
importantly, Gauss’ law and Coulomb’s law are only mathematically
equivalent under the assumption that all our charges are standing
still, and all our fields are constant over time, i.e., in the study
of electrostatics, as opposed to electrodynamics. As we broaden
our scope to study generators, inductors, transformers, and radio
antennas, we will encounter cases where Gauss’ law is valid, but
Coulomb’s law is not.10.6.6 Applications
Often we encounter situations where we have a static charge
distribution, and we wish to determine the field. Although super-
position is a generic strategy for solving this type of problem, if the
charge distribution is symmetric in some way, then Gauss’ law is
often a far easier way to carry out the computation.Field of a long line of charge
Consider the field of an infinitely long line of charge, holding a
uniform charge per unit lengthλ. Computing this field by brute-
force superposition was fairly laborious (examples 13 on page 596
and 16 on page 603). With Gauss’ law it becomes a very simple
calculation.
The problem has two types of symmetry. The line of charge,
and therefore the resulting field pattern, look the same if we rotate
them about the line. The second symmetry occurs because the line
is infinite: if we slide the line along its own length, nothing changes.
This sliding symmetry, known as a translation symmetry, tells us
that the field must point directly away from the line at any given
point.
Based on these symmetries, we choose the Gaussian surface
shown in figure i. If we want to know the field at a distanceR
from the line, then we choose this surface to have a radiusR, as648 Chapter 10 Fields