08/28/2013PHY Lecture 011 Light is electromagnetic radiation! = Electric Field = Magnetic Field Assume linear, isotropic, homogeneous media.

08/28/2013PHY Lecture 012 Maxwell’s Equations Published by J.C. Maxwell in 1861 in the paper “On Physical Lines of Force”. Unite classical electricity and magnetism. Predict the propagation of electromagnetic energy away from time varying sources (current and charge) in the form of waves.

08/28/2013PHY Lecture 013 Maxwell’s Equations Four partial differential equations involving E, B that govern ALL electromagnetic phenomena. Gauss’s Law (elec, mag) Faraday’s Law, Ampere’s Law

08/28/2013PHY Lecture 014 Gauss’s Law (elec) Charge density Electric permittivity constant of medium Total charge enclosed Electric charges give rise to electric fields. dS – outward normal E  dS = q

08/28/2013PHY Lecture 015 Gauss’s Law (mag) No Magnetic Monopoles! B  dS = 0

08/28/2013PHY Lecture 016 Faraday’s Law = mag. flux A changing B field gives rise to an E field E field lines close on themselves (form loops) where:

08/28/2013PHY Lecture 017 Ampere’s Law If E const in time: Where: Magnetic permeability of medium Electric currents give rise to B fields. Electric current j = current density

08/28/2013PHY Lecture 018 What Maxwell’s Equations Imply In the absence of sources, all components of E, B satisfy the same (homogeneous) equation: The properties of an e.m. wave (direction of propagation, velocity of propagation, wavelength, frequency) can be determined by examining the solutions to the wave equation.

08/28/2013PHY Lecture 019 What does it mean to satisfy the wave equation? Imaginea disturbance traveling along the x coordinate (1-dim case).

08/28/2013PHY Lecture 0110 What does a wave look like mathematically? General expression for waves traveling in +ve, -ve directions: Argument affects the translation of wave shape. is the velocity of propagation.

08/28/2013PHY Lecture 0111 Waves satisfy the wave equation Try it for f! Use the chain rule, differentiate: This is the (homogeneous) 1-dim wave equation.

08/28/2013PHY Lecture 0112 E, B satisfy the 3-dim wave equation!! can be and

08/28/2013PHY Lecture 0113 Index of Refraction (1) Okay,Velocity of light in a medium dependent on medium’s electric, magnetic properties. In free space:

08/28/2013PHY Lecture 0114 Index of Refraction (2) For any l.i.h. medium, define index of refraction as: NOTE: dimensionless.

08/28/2013PHY Lecture 0115 Index of Refraction (3)

08/28/2013PHY Lecture 0116 Plane waves Back to the 3-dim wave equation, but assume has constant value on planes:

08/28/2013PHY Lecture 0117 Seek solution to wave eqn Solving PDEs is hard, so assume solution of the form: (so-called “separable” solution…) Now, Becomes:

08/28/2013PHY Lecture 0118 Voilà! Two ordinary differential equations! and Note!

08/28/2013PHY Lecture 0119 We know the solutions to these... where. (Sines and cosines!)

08/28/2013PHY Lecture 0120 How to build a wave Choose  positive, +ve z dir, then have Any linear combination of solutions of this form is also a solution. Start with sines and cosines, make whatever shape like.

08/28/2013PHY Lecture 0121 Let’s get physical Sufficient to study wavelength frequency Harmonic wave kz-ωt - phase (radians) ω - angular frequency k – propagation number/vector

08/28/2013PHY Lecture D wave equation Solution: Reduces to the 1-D case when

08/28/2013PHY Lecture 0123 Back to Plane Waves Assume we have (plane waves in the z-direction, E 0 a constant vector) Similar equations for B.,

08/28/2013PHY Lecture 0124 Electromagnetic Waves are Transverse Differentiate first equation of previous slide, can show then using Maxwell’s equations that: Try it!

08/28/2013PHY Lecture 0125 EM Waves are Transverse (2) This implies: Fields must be perpendicular to the propagation direction!

08/28/2013PHY Lecture 0126 EM Waves are Transverse (3) Also, fields are in phase in the absence of sources and E is perpendicular to B since k E B

08/28/2013PHY Lecture 0127 What light looks like close up Magnetic Field Waves Electric Field Waves The Electric and Magnetic components of light are perpendicular (in vacuum). + Moving charge(s) Waves propagate with speed 3x10 8 m/s.

08/28/2013PHY Lecture 0128 The Poynting Vector S is parallel to the propagation direction. In free space, S gives us the energy transport of waveform. Energy/time/area I = time =1/2(c ε 0 ) E irradiance (time average of the magnitude of the Poynting vector)

08/28/2013PHY Lecture 0129 The Electromagnetic Spectrum

08/28/2013PHY Lecture 0130 The Electromagnetic Spectrum