METHOD OF IMAGES
Boundary-value Problems
➢We shall consider practical electrostatic problems where only
electrostatic conditions (charge and potential) at some
boundaries are known and it is desired to find E and V throughout
the region
➢Such problems are usually tackled using:
1. Poisson's equation
2. Or Laplace's equation
3. Or Method of Images
➢These problems are usually referred to as boundary value
problems
Method of Images
➢The method of images, is commonly used to determine V, E, D,
and 𝜌𝑠 due to charges in the presence of conductors
➢By this method, we avoid solving Poisson's or Laplace's equation
but rather utilize the fact that a conducting surface is
equipotential
➢The image theory states that the field due to a charge above a
perfectly conducting plane will remain the same if the
conducting plane is removed and an opposite charge is placed at
a symmetrical location below the plane
Method of Images
➢Examples of point, line, and volume charge configurations are
shown below
A Point Charge Above a Grounded
Conducting Plane
➢Consider a point charge Q placed at a distance h from a perfect
conducting plane of infinite extent as shown in Figure (a)
A Point Charge Above a Grounded
Conducting Plane
➢The image configuration is in Figure (b)
A Point Charge Above a Grounded
Conducting Plane
➢The electric field at point P(x, y, z) is given by:
➢The distance vectors are given as:
➢Therefore:
A Point Charge Above a Grounded
Conducting Plane
➢It should be noted that when z=0, E has only the z-component,
confirming that E is normal to the conducting surface
➢The potential at P can be written as:
A Point Charge Above a Grounded
Conducting Plane
➢The surface charge density of the induced charge can be obtained
as:
➢So the total induced charge on the conducting plane is:
➢By changing variables, 𝜌2 = 𝑥 2 + 𝑦 2 , 𝑑𝑥𝑑𝑦 = 𝜌𝑑𝜌𝑑∅
A Point Charge Above a Grounded
Conducting Plane
➢Therefore:
➢Or:
➢Therefore, all flux lines terminating on the conductor would have
terminated on the image charge if the conductor were absent
A LINE Charge Above a Grounded
Conducting Plane
➢Consider an infinite charge with density 𝜌𝐿 C/m located at a
distance h from the grounded conducting plane z = 0
➢The same image system of point charge applies to the line charge
as well except that Q is replaced by 𝜌𝐿
➢The infinite line charge 𝜌𝐿 may be assumed to be at 𝒙 = 𝟎, 𝒛 =
𝒉 and the image −𝜌𝐿 at 𝒙 = 𝟎, 𝒛 = −𝒉 so that the two are
parallel to the y-axis
➢The electric field at a point P(x,y,z) is given as:
A LINE Charge Above a Grounded
Conducting Plane
➢The distance vectors are given as:
➢So we get:
➢Notice that when z = 0, E has only the z-component, confirming
that E is normal to the conducting surface
A LINE Charge Above a Grounded
Conducting Plane
➢The potential at P is obtained from the line charges as:
➢Substituting the magnitudes of the distance vectors, we get:
A LINE Charge Above a Grounded
Conducting Plane
➢The surface charge induced on the conducting plane is given by:
➢The induced charge per length on the conducting plane is:
➢By letting 𝑥 = ℎ 𝑡𝑎𝑛 ∝, the above equation becomes:
Problem-1
➢A positive point charge Q is located at distance d1 and d2,
respectively from two grounded (V = 0) perpendicular
conducting half planes. Determine the force on charge Q
caused by the charges induced on the planes.
Problem-2
➢Let surface y=0 be a perfect conductor in free space. Two
uniform infinite line charges of 30 nC/m each are located at
x=0, y=1 and x=0, y=2. Let V=0 at the plane y=0, find E at
P(1,2,0)
