Physics 2102 Jonathan Dowling James Clerk Maxwell 1831 1879 Lecture 33 FRI 03 APR Ch 33 1 3 5 E M Waves Maxwell Waves and Light A solution to the Maxwell equations in empty space is a traveling wave d C B ds 0 0 dt S E dA d C E ds dt S B dA electric and magnetic forces can travel d2E d2E 0 0 2 E E0 sin k x ct 2 dx dt c 1 3 10 8 m s 0 0 The electric waves travel at the speed of light Light itself is a wave of electricity and magnetism Electromagnetic Waves A solution to Maxwell s equations in free space E Em sin k x t B Bm sin k x t c speed of propagation k Em c Bm 1 0 0 m 299 462 954 187 163 miles sec s Visible light infrared ultraviolet radio waves X rays Gamma rays are all electromagnetic waves Radio waves are reflected by the layer of the Earth s atmosphere called the ionosphere This allows for transmission between two points which are far from each other on the globe despite the curvature of the earth Marconi s experiment discovered the ionosphere Experts thought he was crazy and this would never work Maxwell s Rainbow The wavelength frequency range in which electromagnetic EM waves light are visible is only a tiny fraction of the entire electromagnetic spectrum Fig 33 2 Fig 33 1 33 2 The Traveling Electromagnetic EM Wave Qualitatively An LC oscillator causes currents to flow sinusoidally which in turn produces oscillating electric and magnetic fields which then propagate through space as EM waves Next slide Fig 33 3 Oscillation Frequency 1 LC 33 3 Mathematical Description of Traveling EM Waves Electric Field E Em sin kx t Magnetic Field B Bm sin kx t Wave Speed c 1 0 0 All EM waves travel a c in vacuum 2 Wavenumber k EM Wave Simulation 2 Angular frequency Vacuum Permittivity 0 Vacuum Permeability 0 Fig 33 5 Amplitude Ratio Em c Bm E t Magnitude Ratio c B t 33 5 The Poynting Vector Points in Direction of Power Flow Electromagnetic waves are able to transport energy from transmitter to receiver example from the Sun to our skin The power transported by the wave and its direction is quantified by the Poynting vector 1 S E B 0 John Henry Poynting 1852 1914 For a wave since E is perpendicular to B In a wave the fields change with time Therefore the Poynting vector changes too Units Watt m2 E S B 1 1 2 S EB E 0 c 0 The direction is constant but the magnitude changes from 0 to a maximum value EM Wave Intensity Energy Density A better measure of the amount of energy in an EM wave is obtained by averaging the Poynting vector over one wave cycle The resulting quantity is called intensity Units are also Watts m2 2 2 2 1 1 I S Em sin kx t E c 0 c 0 1 I Em 2 or 2c 0 Both fields have the same energy density The average of sin2 over one cycle is 1 I Erms 2 c 0 1 1 1 B2 1 B2 2 2 uE 0 E 0 cB 0 uB 2 2 2 0 0 2 0 The total EM energy density is then u 0 E B 0 2 2 Solar Energy The light from the sun has an intensity of about 1kW m2 What would be the total power incident on a roof of dimensions 8m x 20m I 1kW m2 is power per unit area P IA 103 W m2 x 8m x 20m 0 16 MegaWatt The solar panel shown BP275 has dimensions 47in x 29in The incident power is then 880 W The actual solar panel delivers 75W 4 45A at 17V less than 10 efficiency The electric meter on a solar home runs backwards Entergy Pays YOU EM Spherical Waves The intensity of a wave is power per unit area If one has a source that emits isotropically equally in all directions the power emitted by the source pierces a larger and larger sphere as the wave travels outwards 1 r2 Law I Ps 4 r 2 So the power per unit area decreases as the inverse of distance squared Example A radio station transmits a 10 kW signal at a frequency of 100 MHz At a distance of 1km from the antenna find the amplitude of the electric and magnetic field strengths and the energy incident normally on a square plate of side 10cm in 5 minutes Ps 10kW 2 I 0 8 mW m 4 r 2 4 1km 2 1 2 I Em Em 2c 0 I 0 775V m 2c 0 Bm Em c 2 58 nT Received energy P U t S U SAt 2 4 mJ A A
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