Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 387

1
4

1
ε

2

= · − · = − .

4 πε 1

π

2

4. For any two points M 1 , M 2 ∈ Ω so the Green function

is symmetric. While this property can be rigorously proven, we demonstrate it
here physically. Consider an electric field generated by the point electric charge
of capacity

, G ( M 1 , M 2 )= G ( M 2 , M 1 )

1

r at M ,where r is the distance
between M 0 and M .If M 0 is located in a grounded hollow conductor, the electric
potential at M will be a superposition of two parts:

ε at M 0 . The electric potential is

4 π

1

from the point electric

4 π r

charge at M 0 and from the induced electric charges on the inner wall

of the conductor. Since this additional field has no electric charge inside the hol-
low space of the conductor, we have

− g ( M , M 0 )

Δ g = 0 .

Since the conductor is grounded, the electric potential is zero on the surface ∂Ω

of the conductor. Therefore, the g ( M , M 0 ) satisfies

g ∂Ω =

1
4

1

r −

0 r g | ∂Ω =

r . π 4 π

1

( , )= − ( , )

Note that G is the Green function. Therefore, it repre-

sents the electrical potential at M of the electric field generated by a point electric
charge at M 0 inside a grounded hollow conductor. The symmetry of the Green
function agrees with the principle of reciprocity in physics, which states that the
electrical potential at M 2 of an electric field generated by a point electric charge
at M 1 is the same as that at M 1 of the electric field due to a point electric charge
of the same capacity

M M 0 g M M 0

4 π r

ε at M 2 .

Remark 1. If the additional potential due to the induced electric charges is denoted
by g ( M , M 0 ) , the Green function becomes

1

G ( M , M 0 )=

r +

g ( M , M 0 ) .

4 π

Remark 2. In a two-dimensional plane domain, the Green function reads

1
2

ln 1 G ( M , M 0 )=

r −

g ( M , M 0 ) ,

π

where g ( M , M 0 ) satisfies

⎧
⎨

R 2 Δ g = 0 , M ∈ Ω ⊂ ,

1
2

⎩

ln r . g | ∂Ω =

π