tailieunhanh - Ebook Introduction to electrodynamics (3rd edition): Part 2
(BQ) Part 2 book "Griffiths intro to electrodynamics" has content: Conservation laws, electromagnetic waves, potentials and fields, radiation, electrodynamics and relativity. | Chapter 8 Conservation Laws Charge and Energy The Continuity Equation In this chapter we study conservation of energy, momentum, and angular momentum, in electrodynamics. But I want to begin by reviewing the conservation of charge, because it is the paradigm for all conservation laws. What precisely does conservation of charge tell us? That the total charge lP the universe is constant? Well, sure-that's global conservation of charge; but local conservation of charge is a much stronger statement: If the total charge in some volume changes, then exactly that amount of charge must have passed in or out through the surface. The tiger can't simply rematerialize outside the cage; if it got from inside to outside it must have found a hole in the fence. Formally, the charge in a volume V is Q(t) = l per, t) dr, and the current flowing out through the boundary S is charge says dQ dt = _ [ J. da. 1s () Is J . da, so local conservation of () Using Eq. to rewrite the left side, and invoking the divergence theorem on the right, we have [ ap dr = - [ V . J dr, () 1v at 1v and since this is true for any volume, it follows that I~ = () 345 346 CHAPTER 8. CONSERVATION LAWS This is, of course, the continuity equation-the precise mathematical statement of local conservation of charge. As I indicated earlier, it can be derived from Maxwell's equationsconservation of charge is not an independent assumption, but a consequence of the laws of electrodynamics. The purpose of this chapter is to construct the corresponding equations for conservation of energy and conservation of momentum. In the process (and perhaps more important) we willieam how to express the energy density and the momentum density (the analogs to p). as well as the energy "current" and the momentum "current" (analogous to J). Poynting's Theorem In Chapter 2, we found that the work necessary to assemble a static charge distribution (against the Coulomb repulsion
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