Ch 23 Gauss’ Law
1 What is Physics? Gauss’ law relates the electric fields at points on a (closed) Gaussian surface to the
net charge enclosed by that surface.
2 Flux. When a fluid passes through an opening, the rate of volume flow (flux) is given by
⃗ ⃗,
where ⃗ is the fluid velocity and ⃗ is the area vector, which has magnitude equal to the opening surface
area
and is normal to the surface.
⃗ , which
3 Flux of an Electric Field. The flux of an electric field ⃗⃗ through a surface is
∮ ⃗⃗
⃗⃗ ⃗ if appropriate symmetry makes the integrand constant. The electric flux
simplifies to
through a surface is proportional to the net number of electric field lines passing through that surface.



Checkpoint 1
Sample Problem: Flux through a closed cylinder, uniform field
Sample Problem: Flux through a closed tube, non-uniform field
4 Gauss’ Law:
, where
flux though a closed (Gaussian) surface, and
defined as positive, inward as negative.



is the permittivity of free space, is the
is the net charge inside the surface. Outward flux is
Checkpoint 2
Sample Problem: Relating the net enclosed charge and the net flux
Sample Problem: Enclosed charge in a non-uniform field
5 Gauss’ Law and Coulomb’s Law. Application of Gauss’ Law with spherical symmetry to an isolated
point charge results in Coulomb’s law for the electric field,
.

Checkpoint 3
6 A Charged Isolated Conductor. Application of Gauss’ law to a conductor in electrostatic equilibrium
yields a number of simple results. A) If an excess charge is placed on an isolated conductor, that amount
of charge will move entirely to the surface of the conductor. None of the excess charge will be found
within the body of the conductor. B) An isolated conductor with a cavity: There is no net charge on the
cavity walls. (Note: If there is a charge in the space inside the cavity, the conductor is not isolated, and
there is a charge on the cavity wall. C) The conductor removed: The electric field is generated solely by
the excess charges on the conductor surface, and by any external charges. D) The external electric field:
The electric field just outside the surface of a conductor is perpendicular to the surface, and its
magnitude is proportional to the (not always uniform) surface charge density , so
.

Sample Problem: Spherical metal shell, electric field and enclosed charge
7 Applying Gauss’ Law: Cylindrical Symmetry. If the charge density has cylindrical symmetry, the field is
radial, pointing away from or toward the (infinite) line charge, and is given by
,
where is the distance from the axis and
is the charge per unit length located within a distance of
the axis.

Sample Problem: Gauss’ Law and an upward steamer in a lightning storm
8 Applying Gauss’ Law: Planar Symmetry
Nonconducting sheet: The electric field generated by an infinite sheet with charge density has
constant magnitude over all space, given by
. It points directly away from (
) or
toward (
) the (infinite) sheet.
Two conducting plates: The field generated by two infinite parallel plates with opposite charge densities
is zero outside the space between the plates. Inside, the field magnitude is
, and the field
points from the positive to the negative plate.

Sample Problem: Electric field near two parallel charged metal plates
9 Applying Gauss’ Law: Spherical Symmetry. A spherical shell of uniform charge attracts or repels a
charged particle that is outside the shell as if all the shell’s charge were concentrated at the center of
the shell. If a charged particle is located inside a spherical shell of uniform charge, there is no
electrostatic force on the charge from the shell.
In general, if the charge density obeys spherical symmetry, then the field is radial, pointing directly away
from or toward the center of the charge distribution. The field magnitude is
,
where is the radius (distance from the center) and
is the charge within a distance of the center.
For a thin spherical shell with uniformly distributed charge , the field is zero inside the shell, and
follows the usual Coulomb’s Law result,
outside the shell.
If a charge

is uniformly distributed inside a sphere of radius , the field inside
, and by Coulomb’s Law outside .
Checkpoint 4
is given by