Report

Week 9 Maxwell’s Equations Symbols Used E Electric field intensity ρ charge density I B Magnetic flux density ε0 Permittivity current density D Electric displacement μ0 Permeability c speed of light H Magnetic field intensity M Magnetization P Polarization electric current J Demonstrated that electricity, magnetism, and light are all manifestations of the same phenomenon: the electromagnetic field Electric charges generate fields Fields interact with each other Fields act upon charges Electric charges move in space ◦ Charges generate electric fields ◦ Moving charges generate magnetic fields ◦ changing electric field acts like a current, generating vortex of magnetic field ◦ changing magnetic field induces (negative) vortex of electric field ◦ electric force: same direction as electric field ◦ magnetic force: perpendicular both to magnetic field and to velocity of charge Gauss’ Law for Electricity Gauss’ Law for Magnetism Faraday’s Law of Induction Ampere’s Law Integral Form Differential form B and E must obey the same relationship Show that E = Eo cos (ωt - kz) ax satisfies the wave equation Frequency f (cycles per second or Hz) Wavelength λ (meter) Speed of propagation c = f λ Distance (meters) Determine the frequency of an EM wave with a wavelength of ◦ ◦ ◦ ◦ 1000 m (longwave) 30 m (shortwave) 1 cm (microwave) 500 nm (green light) E = Eo cos (ωt - kR) aE H = Ho cos(ωt - kR) aH where A is the amplitude t is time ω is the angular frequency 2πf k is the wave number 2π/λ aE is the direction of the electric field aH is the direction of the magnetic field R is the distance traveled Euler’s Formula A e+jφ = Acos(φ) + jAsin(φ) A cos(φ) = Re {Ae+jφ} Imaginary A sin(φ) = Im {Ae+jφ} Real A e-jφ = A cos(φ) - jA sin(φ) unit circle Show that A cos(φ) = ½ Ae+jφ + ½ Ae-jφ jA sin(φ) = ½ Ae+jφ - ½ Ae-jφ Complex field E = Eo exp (jωt) exp(jψ) aE Phasor convention E = Eo exp(jψ) aE The frequency must be the same The plane wave has a constant value on the plane normal to the direction of propagation The spacing between planes is the wavelength 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. The wave vector is normal to the wave front and its length is the wavenumber |k| = 2π/λ A plane wave propagates in the direction k = 2ax + 1ay + 0.5az Determine the following: ◦ wavelength (m) ◦ frequency (Hz) A plane wave becomes cylindrical when it goes through a slit The wave fronts have the shape of aligned cylinders A spherical wave can be visualized as a series of concentric sphere fronts Poynting Vector (Watts/m2) S = ½ E x H* Poynting Vector (Watts/m2) S = ½ E x H* For plane waves S = |E|2/ 2η Electromagnetic (Intrinsic) Impedance A plane wave propagating in the +x direction is described by E = 1.00 e –jkz ax H = 2.65 e –jkz ay Determine the following: ◦ Direction of propagation ◦ Intrinsic impedance ◦ Power density V/m mA/m Read Chapter Sections 7-1, 7-2, 7-6 Solve Problems 7.1 7.3, 7.25, 7.30, and 7.33