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Divergence and Divergence Theorem
Problem 1:
Vector field E is characterized by the following properties:
a) E points along r ˆ.
b) The magnitude of E is a function of only the distance from the origin.
c) E vanishes at the origin.
d)
E 6,
everywhere.
Find an expression for E that satisfies these properties.
Solution:
E f r r
2 2
2 2
r
r E r f r
r r r r
E
2 2 2 2
r f r 6 r r f r 6 r dr
r
2 3
r f r 2 r C
2
C
f r r
r
Since E vanishes at the origin C 0
f r 2 r
E 2 r r
[V/m]
Divergence and Divergence Theorem
Problem 2:
a) The electric field in a free space region is given by
2 3
E 3 yz x 4 y z y z y z [V/m].
What is the charge density in the region, if any?
b) Use a volume integral to find the total charge in the unit cube with one corner at the
origin, one at (1,0,0), another at (0,1,0) and another at (0,0,1).
c) Check your result in part (b) by using a flux integral to compute the charge in the cube.
Solution:
a)
2
0
V V
D yz z y
2 3
0
8 3 C/m
V
yz z y
b)
1 1 1 1
2 2
0 0
0 0 0 0
Q yz z y dxdydz z z dz
0
C
Q
c)
1 1 1 1
0 0
0 0 0 0
4z C
Q dS dxdz y dxdy
D n
Divergence and Divergence Theorem
Problem 4:
a) Show that the volume charge density corresponding to
2 2
10sin cos
z
D z
is
independent of
and
z.
b) Verify the divergence theorem for D with the cylindrical volume shown below.
x
y
z
a
2
h
2
h
Solution:
a)
z
V
D
D
D
z
D
2 2
10 sin cos
V
z
z
2
2
10sin 10
cos
V
3
C/m.
V
b)
/2 2
/2 0 0
h a
V
h
RHS dV d d dz ah
D
side top bottom
LHS dS dS dS
D n D n D n
/2 2 2
2 2
/2 0 0 0 /
10sin 2 cos
h a
h a z h
z
LHS d dz d d
LHS 10 ah 10 ah 20 ah
Thus, LHS RHS.
Divergence and Divergence Theorem
Problem 5:
A static electric field in free space is given by
2 2
ˆ ˆ
E r ˆ 2 r r sin r sin cos
.
Find the volume charge density that produced this electric field.
Solution:
v 0
D E
In spherical coordinates, we have
2
2
sin
sin sin
r
E
E r E E
r r r r
where
2
r
E r
E r sin
2
E r sin cos
Hence, we have
0
8 2 cos sin [ ]
v
r r
3
C / m .
