–1–
Gauss’s law and Gaussian surface
The integral form of Gauss’s law goes
Z
Z
Z
I
Qenc
1
1
1
~
~
E · da =
=
ρdτ =
σda =
λdl
ǫ0
ǫ0
ǫ0
ǫ0
When the charge distribution (ρ, σ, orλ) is symmetric, we may find a closed surface such that on
a certain part of all of the surface, the electric field is parallel to the surface normal while its
magnitude is constant, and on other parts of the surface, the electric field is perpendicular to the
H
R
R
~ · n̂da = Eda = E da = EA,
surface. In such a situation, the integral can be re-written as E
~ is parallel to the surface normal
where A is the total area of the part of the surface on which E
enc
.
with a constant magnitude E. Using the Gauss’s law, the electric field is thus given by E = QAǫ
Such a surface is called a Gaussian surface.
You may determine the Gaussian surface in the following symmetric configurations and therefore
find the electric field on the Gaussian surface.
• A sphere of radius R with volume charge density ρ(r) as a function of r, the distance from
the center.
• A spherical shell of radius R with uniform surface charge density σ.
• Concentric spherical shells of radius R1 and R2 with uniform surface charge density σ1 and
σ2 .
• An infinitely long line charge with uniform line charge density λ
• An infinitely long cylinder of radius a with volume charge density ρ(s) as a function of s, the
distance from the center axis.
• An infinitely long cylindrical shell of radius a with uniform charge density σ.
• Concentric long cylindrical shells of radii a and b with uniform surface charge density σ1 and
σ2 .
• Infinitely large surface in xy plane with uniform surface charge density σ.
• Infinitely large slab parallel to xy plane with thickness d with volume charge density ρ(z) as
a function of z.
–2–
~ field in symmetric and uniform charge distribution
Table 1. E
geometry
direction
point charge
solid sphere
spherical shell
long line charge
long cylinder
long cylindrical shell
large sheet (in xy plane)
large slab (in xy plane)
r̂
r̂
r̂
ŝ
ŝ
ŝ
±ẑ
±ẑ
magnitude
inside
outside
discontinuity
∝ 1/r 2
∝ 1/r 2
∝ 1/r 2
∝ 1/s
∝ 1/s
∝ 1/s
constant
constant
at the charge
none
on the shell
on the line
none
on the shell
on the sheet
none
∝r
0
∝s
0
∝ |z|