Time variation Combining electrostatics and magnetostatics: (1) .E =  /  o where  =  f +  b (2) .B = 0“no magnetic monopoles” (3)  x E = 0 “conservative” (4)  x B =  o jwhere j = j f + j M Under time-variation: (1) and (2) are unchanged, (3) becomes Faraday’s Law (4) acquires an extra term, plus 3rd component of j

Faraday’s Law of Induction emf  induced in a circuit equals the rate of change of magnetic flux through the circuit dSdS B dℓdℓ

Displacement current Ampere’s Law Problem! Steady current implies constant charge density so Ampere’s law consistent with the Continuity equation for steady currents Ampere’s law inconsistent with the continuity equation (conservation of charge) when charge density time dependent Continuity equation

Extending Ampere’s Law add term to LHS such that taking Div makes LHS also identically equal to zero: The extra term is in the bracket extended Ampere’s Law

Illustration of displacement current C I R S - + oEoE  o  E/  t Discharging capacitor

Displacement current magnitude Suppose E varies harmonically in time  ~ rads -1 for  o  E/  t to be comparable to  E

Types of current j Polarisation current density from oscillation of charges in electric dipoles Magnetisation current density variation in magnitude of magnetic dipoles in space/time M = sin(ay) k k i j j M = curl M = a cos(ay) i Total current

1st form of Maxwell’s Equations all field terms on LHS and all source terms on RHS The sources (  and j) are multiple (free, bound, mag, pol) special status of free source suggests 2nd Form

Extending Ampere’s Law to H  D/  t is displacement current postulated by Maxwell (1862) to exist in the gap of a capacitor being charged In vacuum D =  o E and displacement current exists throughout space

2nd form of Maxwell’s Equations Applies only to well behaved LIH media Focus on sources means equations (2) and (3) unchanged! Recall Gauss’ Law for D In this version of (1),    f and  o   Recall H version of (4) In this version of (4), j  j f, also  o   and  o  

2nd and 3rd forms LHS: 2nd form, free sources only, other sources hidden in permittivity and permeability constants RHS: 3rd form (Minkowsky) free sources only, mixed fields, no constants

Electromagnetic Wave Equation First form

Electromagnetic Waves in Vacuum speed of light in vacuum  = ck = 2  c/  k Dispersion relation

Relationship between E and B E || i B || j k || k

Plane waves in a nutshell r rr r || k Consecutive wave fronts

EM Waves in insulating LIH medium Less than speed of light in vacuum  complex in general, real (as has been assumed) if h <<E g  = ck/n = 2  c/n  k Dispersion relation Slope=±c/n

Bound Charges xBxB vBvB t oo -  /2 -- +  /2 0  xBxB vBvB Phase relative to driving field vs frequency Bound charge displacement x B Or velocity v B versus time

Free Charges For a free charge, spring constant and  o tend to zero

Dielectric susceptibility

EM waves in conducting LIH medium

EM wave is attenuated within ~ skin depth in conducting media NB Insulating materials become ‘conducting’ when radiation frequency tuned above E g

Energy in Electromagnetic Waves Energy density in matter for static fields Average obtained over one cycle of light wave

Energy in Electromagnetic Waves Average energy over one cycle of light wave Distance travelled by light over one cycle = 2  c/  = c  Average energy in volume ab c  a b cc

Energy in Electromagnetic Waves

Poynting Vector N = E x H is the Poynting vector Equal to the instantaneous energy flow associated with an EM wave In vacuum N || wave vector k Example If the E amplitude of a plane wave is 0.1 Vm -1 Energy crossing unit area per second is