METHOD OF IMAGES
Class Activities: Method of Images
3.7
A point charge +Q sits above a very large
grounded conducting slab.
What is E(r) for other points above the slab?
A) Simple Coulomb’s law:
E(r ) =
Â
4pe0 Â3
Q
with  = (r - d ẑ)
B) Something more complicated
+Q
d
r
z
y
x
Infinite grounded conducting
slab
+Q
d
-ẑ- - ŷ- -------- --x̂---+kQ
?
V ( z > 0) =
r - r'
Boundary Conditions: V ( far away) ® 0
V(z = 0) = 0 (grounded)
+Q
d
ẑ
ŷ
x̂
Calculate voltage V(r) (everywhere in space!)
for 2 equal/opposite point charges a distance
“d” above and below the origin.
Where is V(r)=0?
(If you’re done early, figure out Ex, Ey, and Ez
from this voltage, and evaluate (simplify) AT
the plane z=0)
Boundary Conditions: V ( far away) ® 0
V(z = 0) = 0 (grounded)
+Q
d
ẑ
ŷ
x̂
d¢
-Q
Boundary Conditions: V ( far away) ® 0
V(z = 0) = 0 (grounded)
+Q
d
ẑ
-d
ŷ
x̂
-Q
“Image Charge”
+Q
-Q
Method of Images
Ñ V = - r e0
2
Poisson’s Equation
Last Class: Uniqueness Theorem
If: Ñ VA = - r e0
2
& If: VA = VB
Then:
& Ñ2VB = - r e0
on the boundaries
VA = VB everywhere
(within the boundaries)
Boundary Conditions: V ( far away) ® 0
V(z = 0) = 0 (grounded)
+Q
d
V ( r) z³0 = å
i
kQi
r - ri¢
ŷ
x̂
ẑ
-d
-Q
Uniqueness Theorem
Method of Images
Siméon Denis
Poisson
1781-1840
Method of Images
William Thomson
(Lord Kelvin)
1824-1907
Method of Images
James Clerk
Maxwell
1831-1879
“Treatise on Electricity and Magnetism”
1873, Vol. 1, Ch. XI (p. 245-283, 3rd Ed.)
Boundary Conditions: V ( far away) ® 0
V(z = 0) = 0 (grounded)
é
ê
kQ
V ( r) z³0 = ê
2
2
2
êë x + y + ( z - d )
(
-
) (x
1/2
kQ
2
E ( r) = -ÑV ( r)
¶V
s ( x, y) = -e0
¶z
z=0
+ y + (z + d)
2
2
ù
ú
1/2 ú
úû
)
3.8
A point charge +Q sits above a very
large grounded conducting slab. What
is the electric force on this
charge?
2
Q
A) 0
B)
downwards
2
4pe 0 (2d)
C)
Q
2
4pe 0 d
2
downwards
D)
Something
more
complicated
+Q
d
z
y
x
3.8b
A point charge +Q sits above a very
large grounded conducting slab.
What's the energy of this system?
-Q2
A)
4pe0 (2d)
B) Something else.
+Q
dz
y
x
+Q
-Q
3.9
Two ∞ grounded conducting slabs meet
at right angles. How many image
charges are needed to solve for V(r)?
+Q
r
A) one
B) two
C) three
D) more than three
E) Method of images
won't work here
Could we use the method of images
for THIS problem? Find V(r) everywhere z>0,
Given these two charges above a (grounded,
infinite, conducting) plane?
+Q
d
ẑ
A)
B)
C)
D)
Yes, it requires 1 “image charge”
Yes, it requires 2 images
Yes, more than 2 image charge
No, this problem can NOT be
solved using the “trick” of image
charges…
ŷ
x̂
+3Q
2d
Could we use the method of images
for THIS problem? Find V(r) everywhere z>0,
Given these two charges above a (grounded,
infinite, conducting) plane?
+Q
d
ẑ
+3Q
2d
ŷ
x̂
d
-Q
2d
-3Q