Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

−λ

−λ

+λ

+λ

x 0

y 0

r 0

r 2 r 1

r 3

C x

y

V

Figure 21.13 The arrangement of images for finding the force on a line

charge situated in the (two-dimensional) quarter-spacex>0,y>0, when the

planesx=0andy= 0 are earthed.

terms ofx 0 andy 0 , the total force on the line charge due to the charge induced on the
plates is given by

F=

λ^2

2 π 0

(

−

1

2 y 0

j+

2 x 0 i+2y 0 j

4 x^20 +4y^20

−

1

2 x 0

i

)

=−

λ^2

4 π 0 (x^20 +y^20 )

(

y^20

x 0

i+

x^20

y 0

j

)

.

Further generalisations are possible. For instance, solving Poisson’s equation in

the two-dimensional strip−∞<x<∞,0<y<brequires an infinite series of

image points.

So far we have considered problems in which the boundaryS consists of

straight lines (in two dimensions) or planes (in three dimensions), in which simple

reflections of the source atr 0 in these boundaries fix the positions of the image

points. For more complicated (curved) boundaries this is no longer possible, and

finding the appropriate position(s) and strength(s) of the image source(s) requires

further work.

Use the method of images to find the Dirichlet Green’s function for solving Poisson’s

equation outside a sphere of radiusacentred at the origin.

We need to find a solution of Poisson’s equation valid outside the sphere of radiusa.
Since an image pointr 1 cannot lie in this region, it must be located within the sphere. The
Green’s function for this problem is therefore

G(r,r 0 )=−

1

4 π|r−r 0 |

−

q

4 π|r−r 1 |

,

where|r 0 |>a,|r 1 |<aandqis the strength of the image which we have yet to determine.
Clearly,G(r,r 0 )→0 on the surface at infinity.