Gauss s Law Magnetic Gauss s Law Faraday s Law Ampere Maxwell Law Week 14 Day 1 Maxwell s Equation Wave equations Generalization Both electric and magnetic elds travel like waves But there are strict relations between them Concept Question EM Wave Correct answer 2 The electric eld of a plane wave is E z t Esin kz t j The magnetic eld of this wave is given by B z t Bsin kz t Take any function of a single variable y u where u x vt when y x vt ot y x vt or a linear combination is a solution to the one dimensional wave equation y x vt corresponds to a wave traveling in the positive x direction with speed v y x vt corresponds to a wave traveling in the negative x direction with speed v Group Problem Traveling Sine Wave Show that y x t ysin k x vt satis es the one dimensional wave equation Step One differentiate twice with respect to t Step Two differentiate twice with respect to x Wave length and Wave number Spatial Periodicity is the wave length k is the wave number Concept Question Wave Number Correct answer 4 k 2 4 Period Temporal Periodicity The equation for calculating period T is T v Frequency is the inverse of period 1 T v Wavelength and frequency are related by c Angular frequency can be found using the equation 2 T Group Problem One dimensional Sinusoidal EM Waves Consider a plane wave electromagnetic eld given by Show that for the electromagnetic eld to satisfy either condition Show that for the electromagnetic eld to satisfy either wave equation Group Problem Plane Waves Plot E B at each the ten points Why is this a plane wave Concept Question Direction of Propagation Correct answer 4 To nd the direction of propagation do E x B Properties of One Dimensional Waves Concept Question Traveling Wave Poynting Vector and Intensity Correct answer 4 The B eld is in the k hat direction the direction of the electric eld is given by The direction of energy ow is the same as the direction of wave propagation Group Problem Poynting Vector Find the Poynting vector associated with the plane wave Travel with the speed of light At every point in the wave and any instant of time electric and magnetic elds are in phase with one another amplitudes obey E B c Electric and magnetic eld are perpendicular to one another and to the direction of propagation The direction of propagation is the direction of E x B Momentum and Radiation Pressure EM waves transport energy They also transport momentum And exert a pressure This is the only for hitting an absorbing surface For hitting a perfectly re ecting surface the values are doubled as follows
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