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C:\User\Teaching\505506\F10\HW06sol.doc
PHY 505
Classical Electrodynamics I
Fall 2010
Homework 6 with solutions
Problem 6.1
(to be graded of 10 points). An electric dipole is located above an infinite conducting
plane. Calculate:
(i) the distribution of the induced charge in the conductor,
(ii) the force and the torque acting on the dipole, and
(iii) the dipoletoplane interaction energy.
Solutions
:
(i) The problem may be solved by the introduction of a dipole image,
at the same distance
d
below the plane, and with the same dipole moment
magnitude
p
as the original dipole, but reflected in the vertical plane
perpendicular to that containing the dipole moment vector (see Fig. on the
right).
1
Let us prove that. The net field of these two dipoles evidently
satisfies the Poisson equation in the upper halfspace, so that the only thing we have to prove is that it
also satisfies the boundary condition (
= 0) on the plane surface. Let us use Eq. (3.7) of the lecture
notes for of a system of several dipoles – in our case, of two dipoles (let us call them
p
’ and
p
”
), with
Cartesian components
cos
,
0
,
sin
'
'
'
'
'
'
'
'
'
p
p
p
p
p
p
p
p
z
z
y
y
x
x
,
located at points
d
z''
z'
y''
y'
x''
x'
,
0
,
0
.
(Here
x
is the coordinate within the vertical plane which contains vectors
p
’
and
p
”
, i.e. in the plane of
our drawing, while axis
y
is perpendicular to the plane.) In these coordinates, Eq. (3.7) yields
.
)
(
sin
cos
)
(
)
(
sin
cos
)
(
4
)
(
)
(
4
1
2
/
3
2
2
2
2
/
3
2
2
2
0
3
3
0
d
z
y
x
x
d
z
d
z
y
x
x
d
z
p
"
"
"
'
'
'
r

r
p
r
r
r

r
p
r
r
This equation shows that potential
vanishes at an arbitrary point of the surface (
z
= 0), thus proving our
guess.
Now the induced charge may be calculated as
0
0
z
z
,
giving
1
The simplest way to understand this fact is to present the dipole in the form of two point charges (+
q
) and (
q
),
slightly displaced along the direction of the dipole moment vector, and to construct the dipole image from the
mirror images of these point charges in the conducting plane. However, this approach, based on a particular
implementation of a dipole, and can only be used for a
guess
, not as a
proof
.
d
'
p
''
p
d
x
z
0
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