# 4.11 Image Method

Consider a point charge Q at a distance d above a horizontally infinite, perfectly conducting plate (Fig. 4-30(a)). We want to determine V and E at any point in the space above the plate, as well as the surface charge distribution on the plate. Three different methods for finding E have been introduced in this chapter The first method, based on Coulomb’s law, requires knowledge of the magnitudes and locations of all the charges. In the present case, the charge Q induces an unknown and nonuniform distribution of charge on the plate. Hence, we cannot utilize Coulomb’s method. The second method, based on Gauss’s law, is equally difficult to use because it is not clear how to construct a Gaussian surface across which E is only tangential or only normal. The third method is based on evaluating the electric field using E=−∇V after solving Poisson’s or Laplace’s equation for V subject to the available boundary conditions, but it is mathematically involved.

Alternatively, the problem at hand can be solved using image theory.

The image-method equivalent of the charge Q above a conducting plane is shown in the right-hand section of Fig. 4-30. It consists of the charge Q itself and an image charge −Q at a distance 2 d from Q with nothing else between them. The electric field due to the two isolated charges can now be easily found at any point (x, y, z) by applying Coulomb’s method, as demonstrated by Example 4-17. By symmetry, the combination of the two charges always produces a potential V=0 at every point in the plane previously occupied by the conducting surface. If the charge resides in the presence of more than one grounded plane, it is necessary to establish its images relative to each of the planes and then to establish images of each of those images against the remaining planes.

# Figure 4-30 By image theory, a charge Q above a grounded, perfectly conducting plane is equivalent to Q and its image −Q with the grounded plane removed.

Figure 4 30 Full Alternative Text

The process is continued until the condition V=0 is satisfied everywhere on all grounded planes. The image method applies not only to point charges but also to distributions of charge, such as the line and volume distributions depicted in Fig. 4-31. Once E has been determined, the charge induced on the plate can be found from

ρ s=(n ˆ⋅E)ε 0,(4.145)

where n ˆ is the normal unit vector to the plate (Fig. 4-30(a)).

# Figure 4-31 Charge distributions above a conducting plane and their image-method equivalents.

Figure 4 31 Full Alternative Text

# Example 4-17: Image Method for Charge above Conducting Plane

Use image theory to determine E at an arbitrary point P=(x, y, z) in the region z>0 due to a charge Q in free space at a distance d above a grounded conducting plate residing in the z=0 plane.

Solution: In Fig. 4-32, charge Q is at (0,0,d), and its image −Q is at (0,0,−d). From Eq. (4.19), the electric field at point P(x, y, z) due to the two charges is given by the following equation.

E=1 4 π ε 0 Q R 1 R 1 3+−Q R 2 R 2 3=Q 4 π ε 0 x ˆ x+y ˆ y+z ˆ(z−d)[x 2+y 2+(z−d)2]3/2−x ˆ x+y ˆ y+z ˆ(z+d)[x 2+y 2+(z+d)2]3/2

for z≥0.

# Figure 4-32 Application of the image method for finding E at point P (Example 4-17).

Figure 4 32 Full Alternative Text

Concept Question 4-28: What is the fundamental premise of the image method?

Concept Question 4-29: Given a charge distribution, what are the various approaches described in this chapter for computing the electric field E at a given point in space?