Chapter 21: Solutions with Currents & Charges

The foundational discussion begins by examining the electric field produced by a point charge undergoing arbitrary motion. A crucial conceptual element introduced is the retarded time, meaning the electromagnetic field observed at any given moment depends on the charge’s precise position, velocity, and acceleration at an earlier instant. This temporal delay is calculated based on the distance (r) the field must travel at the speed of light (c). The resulting electric field equation includes a component known as the retarded Coulomb field, as well as correction terms that are proportional to the acceleration of the charge. At large distances, these acceleration-dependent terms become dominant, and they are responsible for the production of electromagnetic radiation. The chapter generalizes this process by showing that the scalar potential and vector potential for distributed sources can be found by integrating the effects of all volume elements of charge and current density, incorporating the principle of retardation into these integrals. The phenomenon of radiation is further illustrated through the specific example of an oscillating dipole, confirming that the amplitude of the emitted radiation wave decreases inversely with the distance from the source. Finally, the text addresses the calculation of potentials for a point charge moving at high relativistic velocities. Through rigorous integration and analysis, the necessary Liénard-Wiechert potentials are derived. Applying these potentials to a charge moving at a constant velocity demonstrates a vital connection between Maxwell's equations and the principle of relativity and the Lorentz transformation.