Chapter 27: Field Energy & Momentum in Electromagnetism

Field Energy & Momentum in Electromagnetism comprehensively explores the critical concepts of local conservation of energy and momentum within the framework of electromagnetism, expanding upon the general conservation laws previously introduced. The necessity of local conservation arises because the classical idea of "world-wide" conservation, where physical changes occur instantaneously across distance, violates the principles of relativity. The local conservation law is expressed mathematically as a continuity equation, defining how the rate of change of a density (rho) relates to the divergence of its associated flow (j). Applying this to electromagnetic energy requires defining the field energy density (u), which is the energy stored per unit volume in the electric (E) and magnetic (B) fields. The flow of this energy is described by the energy flow vector, S, historically known as the Poynting vector, which is proportional to the cross product of the electric and magnetic fields and represents the energy flux per unit area per second. Through mathematical manipulation of Maxwell's equations, the energy conservation law is derived, showing that the time derivative of the field energy density is equal to the negative divergence of the Poynting vector. The concept is illustrated with practical examples, such as the flow of energy in a light wave, where S is related to E squared, and the counter-intuitive observation that during the charging of a capacitor, the energy actually flows inward from the surrounding field rather than traveling solely along the conducting wires. The sources acknowledge that while specific derived formulas for u and S exist, there remains an ambiguity in the exact definition of field energy, although the standard formulation is supported because it correctly links energy flow to momentum. Finally, the chapter connects energy flow to field momentum, establishing that momentum is also stored in the electromagnetic field. The momentum density, g, is shown to be proportional to the Poynting vector divided by the speed of light squared (g equals one divided by c squared times S), a relationship confirmed by experiments, like the absorption of light by a moving object. This connection is necessary for the overall conservation of momentum and angular momentum, ensuring that the total momentum of particles plus the field remains constant.