Electrostatics. The Method of Images
1 The Method of Images
The method of images, introduced by Lord Kelvin in 1848, is typically employed to determine
the values of V, E, D, and ρS resulting from charges positioned near conductors. This approach
allows us to avoid directly solving Poisson’s or Laplace’s equations, instead utilizing the prin-
ciple that a conducting surface maintains an equipotential state. Although this technique is not
applicable to all electrostatic problems, it can simplify a complex issue into a more manage-
able one.
According to this theory, if we have a charge configuration above an infinite conducting
plane. Then the charge configuration in the presence of the conducting plane may be replaced
by the charge configuration itself, an equipotential surface in the place of the conducting plane
and an image of the original charge configuration.
Figure 1
1
, The following conditions must be satisfied to apply the method of images.
• The distribution of image charges must be located within the conducting region. This
requirement guarantees that Poisson’s equation is fulfilled.
• The placement of the image charges must ensure that the potential on the conducting
surface remains either zero or constant. This requirement is essential for fulfilling the
boundary condition.
2 Examples
Below, a few straightforward examples illustrating the application of the method of images
will be examined.
2.1 Point Charge Above a Grounded Conducting Plane
Figure 2
The position vector of the point P is
~ = ρ ρ̂ + z ẑ
OP (1)
in cylindrical co-ordinatres.
~ = h ẑ
OQ (2)
2
1 The Method of Images
The method of images, introduced by Lord Kelvin in 1848, is typically employed to determine
the values of V, E, D, and ρS resulting from charges positioned near conductors. This approach
allows us to avoid directly solving Poisson’s or Laplace’s equations, instead utilizing the prin-
ciple that a conducting surface maintains an equipotential state. Although this technique is not
applicable to all electrostatic problems, it can simplify a complex issue into a more manage-
able one.
According to this theory, if we have a charge configuration above an infinite conducting
plane. Then the charge configuration in the presence of the conducting plane may be replaced
by the charge configuration itself, an equipotential surface in the place of the conducting plane
and an image of the original charge configuration.
Figure 1
1
, The following conditions must be satisfied to apply the method of images.
• The distribution of image charges must be located within the conducting region. This
requirement guarantees that Poisson’s equation is fulfilled.
• The placement of the image charges must ensure that the potential on the conducting
surface remains either zero or constant. This requirement is essential for fulfilling the
boundary condition.
2 Examples
Below, a few straightforward examples illustrating the application of the method of images
will be examined.
2.1 Point Charge Above a Grounded Conducting Plane
Figure 2
The position vector of the point P is
~ = ρ ρ̂ + z ẑ
OP (1)
in cylindrical co-ordinatres.
~ = h ẑ
OQ (2)
2
