ABSTRACT
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, ρ V = ρ V t − R S P c . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn14g_1.jpg"/> 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 Φ P r , t = Q t − R S P c 4 π ε 0 R S P , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn14g_2.jpg"/> where R S P = r − r ′ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn14g_3.jpg"/> 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 a dipole source. A single charge is used here mainly to introduce the concept of retarded time. Even if Q is not moving, we have to use the value of the charge at a retarded time t r : t r = t − R S P c . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn14g_4.jpg"/> Let us now consider a point charge moving with a velocity v along a specified trajectory as shown in Figure 14G.1
