In this sense, the terms electric and magnetic “ﬂux densities” for the quantities D , B are somewhat of a misnomer because they do not represent anything that ﬂows. 10 1. Maxwell’s Equations Fig. 1.6.1 Flux of a quantity. Similarly, when J represents momentum ﬂux, we expect to have J mom = cρ mom . Momentum ﬂux is defined as J mom = Δp/(ΔSΔt) = ΔF/ΔS , where p denotes momen- tum and ΔF = Δp/Δt is the rate of change of momentum, or the force, exerted on the surface ΔS . Thus, J mom represents force per unit area, or pressure. Electromagnetic waves incident on material surfaces exert pressure (known as ra- diation pressure), which can be calculated from the momentum ﬂux vector. It can be shown that the momentum ﬂux is numerically equal to the energy density of a wave, that is, J mom = ρ en , which implies that ρ en = ρ mom c . This is consistent with the theory of relativity, which states that the energy-momentum relationship for a photon is E = pc . 1.7 Charge Conservation Maxwell added the displacement current term to Amp` ere’s law in order to guarantee charge conservation. Indeed, taking the divergence of both sides of Amp` ere’s law and using Gauss’s law ∇ · D = ρ , we get: ∇ · ∇ × H = ∇ · J + ∇ · ∂ D ∂t = ∇ · J + ∂ ∂t ∇ · D = ∇ · J + ∂ρ ∂t Using the vector identity ∇· ∇× H = 0, we obtain the differential form of the charge conservation law: ∂ρ ∂t + ∇ · J = 0 (charge conservation) (1.7.1) Integrating both sides over a closed volume V surrounded by the surface S , as shown in Fig. 1.7.1, and using the divergence theorem, we obtain the integrated form of Eq. (1.7.1): S J · d S = − d dt V ρ dV (1.7.2) The left-hand side represents the total amount of charge ﬂowing outwards through the surface S per unit time. The right-hand side represents the amount by which the charge is decreasing inside the volume V per unit time. In other words, charge does not disappear into (or created out of) nothingness—it decreases in a region of space only because it ﬂows into other regions. Another consequence of Eq. (1.7.1) is that in good conductors, there cannot be any accumulated volume charge. Any such charge will quickly move to the conductor’s surface and distribute itself such that to make the surface into an equipotential surface.
1.8. Energy Flux and Energy Conservation 11 Fig. 1.7.1 Flux outwards through surface. Assuming that inside the conductor we have D = E and J = σ E , we obtain ∇ · J = σ ∇ · E = σ ∇ · D = σ ρ ∂ρ ∂t + σ ρ = 0 (1.7.3) with solution: ρ( r , t) = ρ 0 ( r )e − σt/ where ρ 0 ( r ) is the initial volume charge distribution. The solution shows that the vol- ume charge disappears from inside and therefore it must accumulate on the surface of the conductor. The “relaxation” time constant τ rel = /σ is extremely short for good conductors. For example, in copper, τ rel = σ = 8 .
You've reached the end of your free preview.
Want to read all 18 pages?
- Spring '14
- Energy, Electric charge, Permittivity, Dielectric, unit volume