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Electromagnetic Waves, & Their Speed, Derived from Maxwell’s Equations

Electromagnetic Waves, & Their Speed, Derived from Maxwell’s Equations. The Differential (Local) Form of Maxwell’s Equations. Electromagnetic Waves, & Their Speed, Derived from Maxwell’s Equations. The Differential (Local) Form of Maxwell’s Equations.

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Electromagnetic Waves, & Their Speed, Derived from Maxwell’s Equations

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  1. Electromagnetic Waves, & Their Speed, Derived from Maxwell’s Equations The Differential (Local) Form of Maxwell’s Equations

  2. Electromagnetic Waves, & Their Speed, Derived from Maxwell’s Equations The Differential (Local) Form of Maxwell’s Equations

  3. Electromagnetic Waves, & Their Speed, Derived from Maxwell’s Equations In the absence of currents and charges, the integral forms of Maxwell’s equations become: The Integral (Global) Form of Maxwell’s Equations

  4. . This figure shows an electromagnetic wave of wavelength λ & frequency f. The electric & magnetic fields are given by: where

  5. . . Applying Faraday’s law to the rectangle of height Δy & width dx gives a relationship between E & B: Similarly, applying Maxwell’s 4th equation to the rectangle of length Δz & width dx, gives

  6. . Using these two equations and the equations for B and E as a function of time gives Here, v is the velocity of the wave. Substituting, The magnitude of this speed is 3.0  108 m/s – precisely equal to the measured speed of light.

  7. Example:Determining E and B in EM waves. Assume a 60-Hz EM wave is a sinusoidal wave propagating in the z direction with E pointing in the x direction, and E0 = 2.0 V/m. Write vector expressions for E and B as functions of position and time.

  8. Light as an Electromagnetic Wave and the Electromagnetic Spectrum The frequency of an electromagnetic wave is related to its wavelength and to the speed of light: Electromagnetic waves can have any wavelength; we have given different names to different parts of the wavelength spectrum.

  9. Example: Wavelengths of EM waves. Calculate the wavelength (a) of a 60-Hz EM wave, (b) of a 93.3-MHz FM radio wave, and (c) of a beam of visible red light from a laser at frequency 4.74 x 1014 Hz. Example: Cell phone antenna. The antenna of a cell phone is often ¼ wavelength long. A particular cell phone has an 8.5-cm-long straight rod for its antenna. Estimate the operating frequency of this phone.

  10. Example: Phone call time lag. You make a telephone call from New York to a friend in London. Estimate how long it will take the electrical signal generated by your voice to reach London, assuming the signal is (a) carried on a telephone cable under the Atlantic Ocean, and (b) sent via satellite 36,000 km above the ocean. Would this cause a noticeable delay in either case?

  11. Measuring the Speed of Light The speed of light was known to be very large, although careful studies of the orbits of Jupiter’s moons showed that it is finite. One important measurement, by Michelson, used a rotating mirror: Over the years, measurements have become more and more precise; now the speed of light is defined to be c = 2.99792458 × 108 m/s. This is then used to define the meter.

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