Method of images for line capacitance

The method of images is a technique for finding the electric field (and hence capacitance) of charged conductors above a conducting plane. The earth is treated as a perfectly conducting, zero-potential plane. Its effect is replaced by imaginary "image" conductors placed as far below the ground as the real conductors are above it, carrying equal and opposite charges.

Basis

• The earth's surface is an equipotential (zero potential) surface.
• A charge $+q$ at height $h$ and a charge $-q$ at depth $h$ below the surface together produce zero potential everywhere on the plane midway between them.
• So, for the region above ground, the field of "conductor + earth" is the same as the field of "conductor + image conductor" with the earth removed.

     +q o  a           +q o  b      (real conductors)
        |  h              |
 ====================================  earth (V = 0)
        |  h              |
     -q o  a'          -q o  b'     (images)

Use in capacitance calculation

1. Replace the earth by image conductors: for each conductor of charge $q$ at height $h$, add an image of charge $-q$ at depth $h$.
2. Write the voltage of each conductor due to all real and image charges, using $V = \dfrac{q}{2\pi\varepsilon_0}\ln\dfrac{D_2}{D_1}$ for each charge.
3. For a single-phase line (conductors a, b, spacing $D$, height $h$), this gives

$$V_{ab} = \frac{q}{\pi\varepsilon_0}\ln\frac{2hD}{r\sqrt{4h^2+D^2}}$$ $$C_{ab} = \frac{\pi\varepsilon_0}{\ln\left(\dfrac{D}{r\sqrt{1+D^2/4h^2}}\right)}\ \text{F/m}$$

4. For a transposed three-phase line, average over the transposition cycle:

$$C_n = \frac{2\pi\varepsilon_0}{\ln\dfrac{D_{eq}}{r} - \ln\dfrac{\sqrt[3]{H_{12}H_{23}H_{31}}}{\sqrt[3]{H_1H_2H_3}}}\ \text{F/m}$$

where $H_1, H_2, H_3$ are the distances from each conductor to its own image and $H_{12}$ etc. are the distances to the images of the other conductors.

Effect

The extra term subtracted in the denominator is positive, so the denominator becomes smaller and the ground increases the line capacitance slightly. The effect is small when the conductors are high above ground compared with their spacing, so it is usually neglected except for low lines or accurate studies.