MasteringPhysics: Assignment Print View
Gauss's Law in 3, 2, and 1 Dimension
Gauss's law relates the electric flux
through a closed surface to the total charge
enclosed
by the surface:
.
You can use Gauss's law to determine the charge enclosed inside a closed surface on which the
electric field is known. However, Gauss's law is most frequently used to determine the electric field
from a symmetric charge distribution.
The simplest case in which Gauss's law can be used to determine the electric field is that in which
the charge is localized at a point, a line, or a plane. When the charge is localized at a point, so that
the electric field radiates in three-dimensional space, the Gaussian surface is a sphere, and
computations can be done in spherical coordinates. Now consider extending all elements of the
problem (charge, Gaussian surface, boundary conditions) infinitely along some direction, say along
the
z
axis. In this case, the point has been extended to a line, namely, the
z
axis, and the resulting
electric field has cylindrical symmetry. Consequently, the problem reduces to two dimensions, since
the field varies only with
x
and
y
, or with
and
in cylindrical coordinates. A one-dimensional
problem may be achieved by extending the problem uniformly in two directions. In this case, the
point is extended to a plane, and consequently, it has planar symmetry.
Consider a point charge
in three-dimensional space. Symmetry requires the electric field to point
directly away from the charge in all directions. To find
, the magnitude of the field at distance
from the charge, the logical Gaussian surface is a sphere centered at the charge. The electric field
is normal to this surface, so the dot product of the electric field and an infinitesimal surface element
involves
. The flux integral is therefore reduced to
, where
is the magnitude of the electric field on the Gaussian surface, and
is the area of the
surface.