Electricity and Magnetism Unit E5: Maxwell's Equations and Electromagnetic Waves
Prerequisite: E4, W1
Maxwell's modification of Ampere's Law is an example of a new physical law found, not from experiment, but from carefully analyzing the inconsistency of the existing laws and finding a way to modify them so as to make them consistent. The modified laws had important consequences, which were verified by experiment. Most new physical laws are not found in this way, but when they are it is a triumph of human intellect.
Section I: Maxwell's Laws
E. M. Purcell, Electricity and Magnetism, Berkeley Physics Course, Vol. 2, 2nd Ed. , Chapt. 9 - Maxwell's Equations and Electromagnetic Waves, Sec. 9.1-9.3.
Maxwell's equations above are written in their "integral forms". You will probably find it useful to look at the equivalent "differential forms" in Eqn. 15 on page 330 of Purcell. In these equations the symbols "curl" and "div" are both linear in the three partial differential operators and operate on vector fields. The curl acts like a vector cross product and operating on a vector field C(r) produces another vector field: curl C = i(partial y C_z - partial z C_y ) + j (partial z C_x - partial x C_z ) + k(partial x C_y - partial y C_x ), where i, j and k are unit vectors in the x, y and z directions. In contrast the divergence (div) acts like a dot product and operating on a vector field produces a scalar field: div C(r) = partial x C_x + partial y C_y + partial z C_z. Of the four Maxwell's equations, then, the two curl equations are vector equations while the two divergence equations are scalar equations.
Section II: Electromagnetic Waves
E. M. Purcell, Electricity and Magnetism, Berkeley Physics Course, Vol. 2, 2nd Ed. , Chapt. 9 - Maxwell's Equations and Electromagnetic Waves, Sec. 9.4-9.6.
Purcell 9.1, 9.2, 9.5, 9.10 .