Chapter 18: Maxwell’s Equations – Complete Explanation

Maxwell’s Equations – Complete Explanation details the profound synthesis of classical physics laws represented by Maxwell's equations, presented alongside the conservation of charge, the Lorentz force law, and Newton's law of motion. The discussion highlights Maxwell’s crucial contribution: the modification of the fourth equation (the curl of B equation) to incorporate the time rate of change of the electric field (the displacement current term), which was necessary to ensure the universal conservation of electric charge. The necessity of this new term is demonstrated by analyzing scenarios involving localized current flow, such as a spherically symmetric charge distribution and the changing fields inside a charging parallel-plate capacitor. Through the analysis of a simplified physical model—an infinite sheet of charge suddenly set into motion—the mechanism of a traveling electromagnetic field is explored. By solving Maxwell's equations for this propagating disturbance, the text derives conditions relating the electric field (E) and magnetic field (B), proving that these disturbances travel at a unique, constant velocity, which equals c. A historical measurement revealed that the constant formed by the permittivity and permeability of free space yields the inverse square of the speed of light, establishing the monumental discovery that light is an electromagnetic wave. Such waves are characterized by the E and B fields oscillating perpendicular to each other and to the direction of propagation, maintaining the relationship E equals c times B. To simplify the solution of these coupled equations, the chapter introduces the scalar potential (phi) and the vector potential (A) to define E and B. Finally, employing the Lorentz gauge condition (which relates the divergence of A to the time derivative of phi) allows for the decoupling and simplification of the potential equations into familiar wave equations, definitively confirming the propagation of electromagnetic energy at the speed of light c.