79
Chapter 22
The Electric Field 2: Continuous Charge
Distributions
Conceptual Problems
*1
••
(
a
)
False.
Gauss’s law states that the net flux through any surface is given
by
inside
S
n
net
4
kQ
dA
E
π
φ
=
=
∫
. While it is true that Gauss’s law is easiest to apply to
symmetric charge distributions, it holds for
any
surface.
(
b
)
True
2
••
Determine the Concept
Gauss’s law states that the net flux through any surface is given
by
inside
S
n
net
4
kQ
dA
E
π
φ
=
=
∫
. To use Gauss’s law the system must display some
symmetry.
3
•••
Determine the Concept
The electric field is that due to all the charges, inside and
outside the surface. Gauss’s law states that the net flux through any surface is given
by
inside
S
n
net
4
kQ
dA
E
π
φ
=
=
∫
. The lines of flux through a Gaussian surface begin on
charges on one side of the surface and terminate on charges on the other side of the
surface.
4
••
Picture the Problem
We can show that the charge inside a sphere of radius
r
is
proportional to
r
3
and that the area of a sphere is proportional to
r
2
. Using Gauss’s law,
we can show that the field must be proportional to
r
3
/
r
2
=
r
.
Use Gauss’s law to express the
electric field inside a spherical
charge distribution of constant
volume charge density:
A
kQ
E
inside
4
π
=
where
2
4
r
A
π
=
.
Express
Q
inside
as a function of
ρ
and
r
:
3
3
4
inside
r
V
Q
πρ
ρ
=
=
Substitute to obtain:
r
k
r
r
k
E
3
4
4
4
2
3
3
4
πρ
π
πρ
π
=
=

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