1. Maxwell’s Equations and Plane Wave
𝛻× 𝐻 = 𝐽 + 𝜕 𝐷 𝜕𝑡
𝛻∙ 𝐷 =𝜌
𝛻∙ 𝐵 =0
𝛻× 𝐸 =− 𝜕 𝐵 𝜕𝑡
𝛻 2 𝐸 + 𝛽 2 𝐸 =0
𝛻 2 𝐻 + 𝛽 2 𝐻 =0
𝐽 =0, 𝜌=0
𝛻 2 𝐸 = 1 𝑣 2 𝜕 2 𝐸 𝜕 𝑡 2
𝛻 2 𝐻 = 1 𝑣 2 𝜕 2 𝐻 𝜕 𝑡 2
Harmonic
𝐸 𝑟 ,𝑡 = 𝐸 0 cos (𝜔𝑡− 𝛽 ∙ 𝑟 )
𝐻 𝑟 ,𝑡 = 𝐻 0 cos (𝜔𝑡− 𝛽 ∙ 𝑟 ) 𝛽 𝐸 𝐻 𝐸 𝐻 𝛽
Plane wave 𝐸 𝐻 𝛽
𝑣 𝑝 = 𝜔 𝛽
Phase velocity
𝐸 𝐻 = 𝜇 𝜀 =𝜂
Wave impedance
TEM wave

2. An Example Consider the EM wave:
𝐸 =8𝜋 −6 𝑎 𝑥 𝑎 𝑦 cos 2𝜋× 𝑡− 𝜋 𝑥+2𝑦 × V/m
Find:
Field direction (unit vector)
Field amplitude
Propagation direction (unit vector)
Wavenumber and wavelength
Angular frequency and frequency
Phase velocity
Wave impedance
Magnetic field expression
Draw the three directions: propagation, E, and H

3. On 29th August 1831, using his “induction ring”, Faraday made one of his greatest discoveries - electromagnetic induction.