Topics in Classical Electrodynamics

However, the function
1
|⃗x−⃗x ′ | is just one of many functions which obeys ∇2 ψ =
−4πδ (⃗x − ⃗x ′ ). The functions that are solutions of this second order differential
equation are known as Green’s functions. In general,
⃗∇ 2 G (⃗x, ⃗x ′ ) = −4πδ (⃗x − ⃗x ′ ) , (20)
where G (⃗x, ⃗x ′ ) = 1 + F (⃗x, |⃗x−⃗x ′ | ⃗x′ ), so that ∇ ⃗ 2 F (⃗x, ⃗x ′ ) = 0, i.e. it obeys the
Laplace equation inside V . The point is now to find such F (⃗x, ⃗x ′ ), that gets
rid of one of the terms in the integral equation (16) we had for ϕ (⃗x). Letting
ϕ = ϕ (⃗x) and ψ = G (⃗x, ⃗x ′ ), we then get
∫
ϕ (⃗x) =
V
ρ (⃗x ′ ) G (⃗x, ⃗x ′ ) d 3 x ′ + 1
4π
∮
S
[
G (⃗x, ⃗x ′ ) ∂ϕ (⃗x′ )
− ϕ (⃗x ′ ) ∂G (⃗x, ]
⃗x′ )
dS ′ .
∂n ′ ∂n ′
By using the arbitrariness in the definition of the Green function we can leave
in the surface integral the desired boundary conditions. For the Dirichlet case
we can choose G boundary (⃗x, ⃗x ′ ) = 0, when ⃗x ′ ∈ S, then ϕ(⃗x) simplifies to
∫
ϕ (⃗x) =
V
ρ (⃗x ′ ) G (⃗x, ⃗x ′ ) d 3 x ′ − 1
4π
∫
S
ϕ (⃗x ′ ) ∂G (⃗x, ⃗x′ )
∂n ′ dS ′ ,
where G (⃗x, ⃗x ′ ) is referred to as the bulk-to-bulk propagator and ∂G(⃗x,⃗x′ )
is
∂n ′
the bulk-to-boundary propagator.
For the Neumann case we could try to choose ∂G(⃗x,⃗x′ )
= 0 when ⃗x ′ ∈ S.
∂n
However, one has
∮ ∂G (⃗x, ⃗x ′ ∫
)
( ) ∫
∫
dS ′ = ⃗∇G · ⃗n dS ′ = div∇G ∂n ⃗ d 3 x ′ = ∇ 2 G d 3 x ′
′ S
∫
= −4π δ(x − x ′ ) d 3 x ′ = −4π . (21)
For this reason we can not demand ∂G(⃗x,⃗x′ )
= 0. Instead, one chooses another
∂n
simple condition ∂G(⃗x,⃗x′ )
= − 4π , where S is the total surface area, and the
∂n
S
left hand side of the equation is referred to as the Neumann Green function.
Using this condition:
∫
ϕ (⃗x) = ρ (⃗x ′ ) G N (x, x ′ ) d 3 x ′ + 1 ∮
G N (⃗x, ⃗x ′ ) ∂ϕ (⃗x′ )
dS ′
V
4π S
∂n ′
+ 1 ∫
ϕ (⃗x ′ ) dS ′ (22)
S
S
14