Principles of Electromagnetic Waves and Materials, 2nd Edition

Appendix 14G: Moving Point Charge and Lienard–Wiechert Potentials

In Appendix 1B, we wrote down the scalar wave equation (1B.30) and its solution (1B.31) in terms of the integral of the charge density at the retarded time [ρ_V]. For the free space medium,

(14G.1)$$\left[{\mathrm{\rho }}_V\right]={\mathrm{\rho }}_V\left(t-\frac{R_{SP}}{c}\right).$$

Assuming that a point charge Q is defined as an integral of the volume charge density in the limit the volume shrinks to a point, the solution for the scalar potential can be written as

(14G.2)$${\mathrm{\Phi }}_P\left(r\text{,}t\right)=\frac{Q\left(t-\frac{R_{SP}}{c}\right)}{4\mathrm{\pi }{\mathrm{\epsilon }}_0{R}_{SP}},$$

where

(14G.3)$$R_{SP}=\left|r-r^{\prime }\right|.$$

In the above, S is the source point where Q(t) is permanently located and P is the field point. This cannot be an isolated, time-varying, single charge because of the conservation of charge requirement and can be one of the charges in ...