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

An Example Consider the EM wave: 𝐸 =8𝜋 −6 𝑎 𝑥 +3 5 𝑎 𝑦 cos 2𝜋× 10 15 𝑡− 𝜋 3 5 𝑥+2𝑦 × 10 7 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

On 29th August 1831, using his “induction ring”, Faraday made one of his greatest discoveries - electromagnetic induction. https://www.theiet.org/resources/library/archives/biographies/faraday.cfm