Page 337 - Intro to Tensor Calculus

331

where ˆ n is a unit normal to the surface. The flux φ E can be thought of as being proportional to the number
of electric field lines passing through an element of surface area. If the surface is a closed surface we have
by the divergence theorem of Gauss
ZZZ ZZ
~
~
φ E = ∇· Edτ = E · ˆ n dσ
V S
where V is the volume enclosed by S.

Gauss Law
Let dσ denote an element of surface area on a surface S. A cone is formed if all points on the boundary
of dσ are connected by straight lines to the origin. The cone need not be a right circular cone. The situation
is illustrated in the figure 2.6-3.

Figure 2.6-3. Solid angle subtended by element of area.

We let ~ denote a position vector from the origin to a point on the boundary of dσ and let ˆ n denote a
r
r
r
unit outward normal to the surface at this point. We then have ˆ n · ~ = r cos θ where r = |~| and θ is the
r
angle between the vectors ˆ n and ~. Construct a sphere, centered at the origin, having radius r. This sphere
dΩ
intersects the cone in an element of area dΩ. The solid angle subtended by dσ is defined as dω = . Note
r 2
that this is equivalent to constructing a unit sphere at the origin which intersect the cone in an element of
area dω. Solid angles are measured in steradians. The total solid angle about a point equals the area of the
sphere divided by its radius squared or 4π steradians. The element of area dΩ is the projection of dσ on the
ˆ n · ~r ˆ n · ~r dΩ
constructed sphere and dΩ= dσ cos θ = dσ so that dω = dσ = . Observe that sometimes the
r r 3 r 2
dot product ˆ n · ~ is negative, the sign depending upon which of the normals to the surface is constructed.
r
(i.e. the inner or outer normal.)
The Gauss law for electrostatics in a vacuum states that the flux through any surface enclosing many
charges is the total charge enclosed by the surface divided by 0 . The Gauss law is written
ZZ
Q e for charges inside S
~
E · ˆ n dσ = 0 (2.6.21)
S 0 for charges outside S