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Physics 8B
Professor Catherine Bordel
09/08/10
Lecture 5
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LECTURE
Gauss’s Law – the electric flux,
Φ
E
, through a
closed surface (s) is equal to the total amount of
electric charge Q
enc
enclosed in the closed surface
(s) divided by the permittivity of vacuum
ε
0
.
∫
=
⋅
=
)
(
0
s
enc
E
Q
A
d
E
ε
φ
v
v
The circle through the integral symbol means that
the integral is taken over a closed surface. This
surface is a figurative surface, meaning it does not
exist and does not carry electric charge. Gauss’s
Law is one of the four fundamental laws of
electromagnetism. I’d like to cover an example
right now with you.
We’re going to consider a hollow sphere with a
charged surface. The center is O, and the radius is
R. You have a charge distribution on the surface
that is
σ
, and we assume
σ
is positive. If it’s
negative, then E will point in the opposite direction,
but it has the same symmetry and doesn’t change
the way you need to solve the problem.
We want to
calculate the electric field at point M, which is a
distance r away from the center O. You see that the
physical surface that carries the electric charge is
the natural boundary between two regions of space,
the region that is enclosed by the sphere and the
region outside. We will have to study distances of
r>R and also r<R. In each case we need to calculate
the flux going through the closed surface that we
choose, and we will be able to determine the
electric field.
The first step you have to do to use Gauss’s Law is
to figure out by the symmetry of the charge
distribution what the direction of the electric field is
going to be. What is going to be the direction of the
electric field from this spherical charge
distribution?
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