Energy Density and the Poynting Vector Overview and Motivation: We saw in the last lecture that electromagnetic waves are one consequence of Maxwell's (M's) equations. We introduce electromagnetic po-tentials, and show how they can be used to simplify the calculation of the fields in the presence of sources. With electromagnetic waves, as with other waves, there is an associated energy density and energy flux. The power flows with a density S (watts/m2), a vector, so that the power crossing a surface Sa is given by Sa gation of electromagnetic fields as waves. Consider an inertial frame in which the 3-velocity field of the particles is . Let there be particles per unit proper volume (unit volume determined in the local rest frame), each carrying a charge . The electromagnetic energy tensor Consider a continuous volume distribution of charged matter in the presence of an electromagnetic field. H = 0 ∇×E = −(1/c)(∂H/∂t) ∇×H = (1/c)(∂E/∂t) + (4π/c)j where c is the speed of light in a vacuum, ρ is charge density and j is current density. dissipation. Here we introduce these electromagnetic quantities and discuss the conservation of energy in the electromagnetic … The energy density is positive-definite: ≥ The symmetry of the tensor is as for a general stress–energy tensor in general relativity.The trace of the energy-momentum tensor is a Lorentz scalar; the electromagnetic field (and in particular electromagnetic waves) has no Lorentz-invariant energy scale, so its energy-momentum tensor must have a vanishing trace. ... We have completed the derivation of the energy densities for the Dude model, the wire-SRR metamaterial, and the single-resonance chiral metamaterial. 5.6 Electromagnetic Power Density 5.6.1 Poynting Vector Consider Maxwell’s equations in the time domain modified as follows: H ¢r£E = ¡H ¢ @B @t (5.150) E ¢r£H = E ¢ @D @t +E ¢J (5.151) The divergence of a cross product can be expanded using the identity r¢(E £H) = H¢r£E ¡E ¢r£H, which is analogous to the product rule for the scalar derivative. An important problem about dispersive media is how to identify the electromagnetic energy density in them [1, 4, 5]. In a field, theoretical generalization, the energy must be imagined dis­ tributed through space with an energy density W (joules/m3), and the power is dissipated at a local rate of dissipation per unit volume Pd (watts/m3). Our method is based on the Lagrangian dynamics. We derive Poynting’s theorem, which leads to ex-pressions for the energy density and energy flux in an electromagnetic field. Electromagnetic energy is the energy that comes from electromagnetic radiation, such as radio waves and visible light waves, which triggers both electric and magnetic fields.