Lecture 34: MON 13 APR Ch.33.1–3,5–7: E&M WavesPowerPoint PresentationSlide 3Maxwell, Waves and LightElectromagnetic WavesSlide 6Slide 7Slide 8Maxwell’s RainbowSlide 10Slide 11The Poynting Vector: Points in Direction of Power FlowEM Wave Intensity, Energy DensitySolar EnergyEM Spherical WavesExampleRadiation PressureSlide 18Slide 19EM waves: polarizationEM Waves: PolarizationSlide 22Slide 23Lecture 34: MON 13 APRLecture 34: MON 13 APR Ch.33.1–3,5–7: E&M Waves Ch.33.1–3,5–7: E&M WavesJames Clerk Maxwell (1831-1879) Physics 2102Jonathan DowlingQuickTime™ and a decompressorare needed to see this picture.MT03 Avg: 65/100Q1/P3 K. Schafer Office hours: MW 1:30-2:30 pm 222B NicholsonP1/Q2 J. Dowling Office hours: MWF 10:30-11:30 am 453 NicholsonP2/Q3 M. Gaarde Office hours: TTh 2:30-3:30 pm 215B NicholsonP3/Q2 C. Buth Office hours: MF 2:30-3:30 pm 222A NicholsonA: 90-100 B: 80-89 C: 60-79 D: 50-59Problem 1 [18 points] In the figure below, two semicircular arcs (I & II) have radii R1 = 4.10 cm and R2 = 6.60 cm, carry current i = 0.281 A, and share the same center of curvature C. The same current i also flows through the straight sections of wire labeled III & IV. (a) (5 pts) What is the contribution to the magnitude of the magnetic field at point C from the two straight sections of wire III and IV? Explain your answer! (b) (8 pts) Calculate the magnitude of the magnetic field at the point C due to all four sections of wire. (c) (5 pts) What is the direction of the total magnetic field at the point C due to all four sections of wire? Circle one: Out of the page )(•. Into the page ⊗. Up towards the top of the page ↑. Down towards the bottom of the page ↓. To the right of the page →. To the left of the page ←. The total magnetic feld at C is zero and has no direction. BIII=BIV=0drB ∝ drs ×rr =0Biot-Savart BTOT=rBI−rBII=μ0iϕ4π1RI−1RII⎛⎝⎜⎞⎠⎟=8.14×10−7Tϕ =π =half a circleRight Hand Rule & BI>BII Since I is Closer!Question 2 [8 points] The figure below shows four arrangements in which long parallel wires carry equal currents i directly into or out of the page at the corners of identical squares. (i) (4 pts) For square B, what is the direction of the magnetic field, with respect to the page, at the center of the square? Circle one: Out )(• In ⊗ Up ↑ Down↓ Left ← Right → Field is Zero )0(=Br (ii) (4 pts) Rank the sections according to the magnitude of the magnitude of the magnetic feld at the center of each square, greatest frst. Circle one: BA > BB > BC BC > BB > BA BA > BC > BB BB > BC > BA BA = BB = BC Right Hand Rule & Vector Addition!B=μ0i2πr@;@@@@@@BA=0;@@@@@BB=2 2B;@@@@@BC=2B;A solution to the Maxwell equations in empty space is a “traveling wave”…c =1μ0ε0=3×108m/sc =1μ0ε0=3×108m/sThe “electric” waves travelat the speed of light!Light itself is a wave of electricity and magnetism!Maxwell, Waves and LightMaxwell, Waves and Lightd2Edx2=−μ0ε0d2Edt2⇒ E =E0sink(x−ct)d2Edx2=−μ0ε0d2Edt2⇒ E =E0sink(x−ct)electric and magnetic “forces” can travel! ∫∫•=•SCdAEdtddsB00εμ∫∫•=•SCdAEdtddsB00εμ∫∫•−=•SCdABdtddsE∫∫•−=•SCdABdtddsEA solution to Maxwell’s equations in free space:)sin( txkEEmω−=)sin( txkBBmω−=n.propagatio of speed ,ck=ωc =EmBm=1μ0ε0=299,462,954 ms@=@187,163 miles/secVisible light, infrared, ultraviolet,radio waves, X rays, Gammarays are all electromagnetic waves.Electromagnetic WavesElectromagnetic WavesQuickTime™ and a decompressorare needed to see this picture.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.Fig. 33-1The wavelength/frequency range in which electromagnetic (EM) waves (light) are visible is only a tiny fraction of the entire electromagnetic spectrum.Maxwell’s RainbowFig. 33-2(33-2)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.Fig. 33-3Oscillation Frequency:1LCω =Next slideThe Traveling Electromagnetic (EM) Wave, Qualitatively(33-3)Fig. 33-5Mathematical Description of Traveling EM WavesElectric Field:( )sinmE E kx tω= −Magnetic Field:( )sinmB B kx tω= −Wave Speed:0 01cμ ε=Wavenumber: k =2πλ=ωcFrequency: ω =2πT=2π fVacuum Permittivity:0εVacuum Permeability:0μAll EM waves travel a c in vacuumAmplitude Ratio:mmEcB=Magnitude Ratio:( )( )E tcB t=EM Wave Simulation(33-5)Electromagnetic waves are able to transport energy from transmitterto receiver (example: from the Sun to our skin).The power transported by the wave and itsdirection is quantified by the Poynting vector.John Henry Poynting (1852-1914) 211|| EcEBS00==μμThe Poynting Vector: The Poynting Vector: Points in Direction of Power FlowPoints in Direction of Power FlowEBSUnits: Watt/m2For a wave, sinceE is perpendicular to B: BESrrr×=0μ1In a wave, the fields change with time. Therefore the Poynting vector changes too!! The direction is constant, but the magnitude changes from 0 to a maximum value.I =S =1cμ0___E2=1cμ0Em2sin2(kx−ωt)____________=12cμ0EmThe average of sin2 overone cycle is ½:221mEcI0=μ221mEcI0=μ21rmsEcI0=μ21rmsEcI0=μBoth fields have the same energy density. 2 22 201 1 1 1( )2 2 2 2E BB Bu E cB uε ε εε μ μ0 00 0 0= = = = =or, EM Wave Intensity, Energy EM Wave Intensity, Energy DensityDensityA 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.The total EM energy density is then0220/ με BEu ==0220/ με BEu ==Em2Erms2Solar EnergySolar EnergyThe light from the sun has an intensity of about 1kW/m2. What would be the total power incident on a roof of dimensions 8mHxH20mH?IH=H1kW/m2 is power per unit area.PH=HIAH=H(103 W/m2) x 8m x 20mH=H0.16 MegaWatt!H!The solar panel shown (BP-275) 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 solar panel shown (BP-275) has dimensions 47in
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