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Week 9 Maxwell’s Equations

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Demonstrated that electricity, magnetism, and light are all manifestations of the same phenomenon: the electromagnetic field

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Electric charges generate fields ◦ Charges generate electric fields ◦ Moving charges generate magnetic fields Fields interact with each other ◦ changing electric field acts like a current, generating vortex of magnetic field ◦ changing magnetic field induces (negative) vortex of electric field Fields act upon charges ◦ electric force: same direction as electric field ◦ magnetic force: perpendicular both to magnetic field and to velocity of charge Electric charges move in space

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Gauss’ Law for Electricity Gauss’ Law for Magnetism Faraday’s Law of Induction Ampere’s Law

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Integral Form Differential form

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B and E must obey the same relationship

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Show that E = E o cos (ωt - kz) a x satisfies the wave equation

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Frequency f (cycles per second or Hz) Wavelength λ (meter) Speed of propagation c = f λ Distance (meters)

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Determine the frequency of an EM wave with a wavelength of ◦ 1000 m (longwave) ◦ 30 m (shortwave) ◦ 1 cm (microwave) ◦ 500 nm (green light)

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E = E o cos (ωt - kR) a E H = H o cos(ωt - kR) a H whereA is the amplitude t is time ω is the angular frequency 2πf k is the wave number 2π/λ a E is the direction of the electric field a H is the direction of the magnetic field R is the distance traveled

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Euler’s Formula A e +jφ = Acos(φ) + jAsin(φ) A cos(φ) = Re {Ae +jφ } A sin(φ) = Im {Ae +jφ } A e -jφ = A cos(φ) - jA sin(φ) unit circle Real Imaginary

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Show that A cos(φ) = ½ Ae +jφ + ½ Ae -jφ jA sin(φ) = ½ Ae +jφ - ½ Ae -jφ

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Complex field E = E o exp (jωt) exp(jψ) a E Phasor convention E = E o exp(jψ) a E

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The frequency must be the same

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The plane wave has a constant value on the plane normal to the direction of propagation The spacing between planes is the wavelength

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The magnetic field H is perpendicular to the electric field E The vector product E x H is in the direction of the propagation of the wave.

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The wave vector is normal to the wave front and its length is the wavenumber |k| = 2π/λ

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A plane wave propagates in the direction k = 2a x + 1a y + 0.5a z Determine the following: ◦ wavelength (m) ◦ frequency (Hz)

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A plane wave becomes cylindrical when it goes through a slit The wave fronts have the shape of aligned cylinders

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A spherical wave can be visualized as a series of concentric sphere fronts

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Poynting Vector (Watts/m 2 ) S = ½ E x H*

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Poynting Vector (Watts/m 2 ) S = ½ E x H* For plane waves S = |E| 2 / 2η Electromagnetic (Intrinsic) Impedance

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A plane wave propagating in the +x direction is described by E = 1.00 e –jkz a x V/m H = 2.65 e –jkz a y mA/m Determine the following: ◦ Direction of propagation ◦ Intrinsic impedance ◦ Power density

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Read Chapter Sections 7-1, 7-2, 7-6 Solve Problems , 7.25, 7.30, and 7.33

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