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Maxwell’s Equations

Maxwell’s Equations. The two Gauss’s laws are symmetrical, apart from the absence of the term for magnetic monopoles in Gauss’s law for magnetism

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Maxwell’s Equations

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  1. Maxwell’s Equations • The two Gauss’s laws are symmetrical, apart from the absence of the term for magnetic monopoles in Gauss’s law for magnetism • Faraday’s law and the Ampere-Maxwell law are symmetrical in that the line integrals of E and B around a closed path are related to the rate of change of the respective fluxes

  2. Gauss’s law (electrical): The total electric flux through any closed surface equals the net charge inside that surface divided by eo This relates an electric field to the charge distribution that creates it Gauss’s law (magnetism): The total magnetic flux through any closed surface is zero This says the number of field lines that enter a closed volume must equal the number that leave that volume This implies the magnetic field lines cannot begin or end at any point Isolated magnetic monopoles have not been observed in nature

  3. Faraday’s law of Induction: This describes the creation of an electric field by a changing magnetic flux The law states that the emf, which is the line integral of the electric field around any closed path, equals the rate of change of the magnetic flux through any surface bounded by that path One consequence is the current induced in a conducting loop placed in a time-varying B The Ampere-Maxwell law is a generalization of Ampere’s law It describes the creation of a magnetic field by an electric field and electric currents The line integral of the magnetic field around any closed path is the given sum

  4. Maxwell’s Equation’s in integral form Gauss’s Law Gauss’s Law for Magnetism Faraday’s Law Ampere’s Law

  5. Maxwell’s Equation’s in free space (no charge or current) Gauss’s Law Gauss’s Law for Magnetism Faraday’s Law Ampere’s Law

  6. Hertz’s Experiment • An induction coil is connected to a transmitter • The transmitter consists of two spherical electrodes separated by a narrow gap • The discharge between the electrodes exhibits an oscillatory behavior at a very high frequency • Sparks were induced across the gap of the receiving electrodes when the frequency of the receiver was adjusted to match that of the transmitter • In a series of other experiments, Hertz also showed that the radiation generated by this equipment exhibited wave properties • Interference, diffraction, reflection, refraction and polarization • He also measured the speed of the radiation

  7. Implication • A magnetic field will be produced in empty space if there is a changing electric field. (correction to Ampere) • This magnetic field will be changing. (originally there was none!) • The changing magnetic field will produce an electric field. (Faraday) • This changes the electric field. • This produces a new magnetic field. • This is a change in the magnetic field.

  8. An antenna Hook up an AC source We have changed the magnetic field near the antenna An electric field results! This is the start of a “radiation field.”

  9. Look at the cross section Called: “Electromagnetic Waves” Accelerating electric charges give rise to electromagnetic waves E and B are perpendicular (transverse) We say that the waves are “polarized.” E and B are in phase (peaks and zeros align)

  10. Angular Dependence of Intensity • This shows the angular dependence of the radiation intensity produced by a dipole antenna • The intensity and power radiated are a maximum in a plane that is perpendicular to the antenna and passing through its midpoint • The intensity varies as (sin2θ) / r2

  11. Harmonic Plane Waves At t = 0 l = spatial period or wavelength x l At x = 0 t T = temporal period T

  12. Applying Faraday to radiation

  13. Applying Ampere to radiation

  14. Fields are functions of both position (x) and time (t) Partial derivatives are appropriate This is a wave equation!

  15. The Trial Solution • The simplest solution to the partial differential equations is a sinusoidal wave: • E = Emax cos (kx – ωt) • B = Bmax cos (kx – ωt) • The angular wave number is k = 2π/λ • λ is the wavelength • The angular frequency is ω = 2πƒ • ƒ is the wave frequency

  16. The trial solution

  17. The speed of light (or any other electromagnetic radiation)

  18. The electromagnetic spectrum

  19. Another look

  20. Energy in Waves

  21. Poynting Vector • Poynting vector points in the direction the wave moves • Poynting vector gives the energy passing through a unit area in 1 sec. • Units are Watts/m2

  22. Intensity • The wave intensity, I, is the time average of S (the Poynting vector) over one or more cycles • When the average is taken, the time average of cos2(kx - ωt) = ½ is involved

  23. Radiation Pressure (Absorption of radiation by an object) Maxwell showed: What if the radiation reflects off an object?

  24. Pressure and Momentum • For a perfectly reflecting surface, p = 2U/c and P = 2S/c • For a surface with a reflectivity somewhere between a perfect reflector and a perfect absorber, the momentum delivered to the surface will be somewhere in between U/c and 2U/c • For direct sunlight, the radiation pressure is about 5 x 10-6 N/m2

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