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29. Maxwell’s Equations

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2 Topics Laws of Electric & Magnetic Fields James Clerk Maxwell Maxwell’s Equations

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3 Laws of Electric & Magnetic Fields

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4 James Clerk Maxwell 1831 – 1879 In 1865, Maxwell published a paper entitled: A Dynamical Theory of the Electromagnetic Field, Philosophical Transactions of the Royal Society of London 155, 459-512 (1865). This is one of the greatest scientific papers ever written.

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5 Maxwell’s Equations

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6 Displacement Current Maxwell realized that Ampere’s law is not valid when the current is discontinuous as is true of the current through a parallel plate capacitor: wikimedia.org

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7 Displacement Current He concluded that when the charge within an enclosed surface is changing it is necessary to add to Ampere’s law another current called the displacement current: I D wikimedia.org

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The 2 nd Unification of Forces

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9 0 is the magnetic constant 0 is the electric constant

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10 The 2 nd Unification of Forces From we can write which is the speed of light in vacuum!

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11 Light “We can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.” (1866)

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12 5 th Unification? 4 th Unification? 3 rd Unification 2 nd Unification 1 st Unification

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13 Summary Maxwell’s Equations Gauss’s Law for E Gauss’s Law for B Faraday’s Law Ampere’s Generalized Law

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Electromagnetic Waves

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15 Topics Maxwell’s Wave Equations Waves – Recap Electromagnetic Waves Electromagnetic Radiation

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16 Maxwell’s Wave Equations Wave equation for E Wave equation for B These equations describe electric and magnetic waves traveling in the x direction

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17 Maxwell’s Wave Equations Relationship between E z and B y Relationship between B z and E y Maxwell showed that the different components of the electric and magnetic fields are related:

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18 Waves – Recap Stationary wave Wave traveling in x direction Wave number

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19 Electromagnetic Waves Consider an electric wave, traveling in the positive x direction, but oscillating in the y direction: We can find B z from EpEp

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20 Electromagnetic Waves This leads to the result where BpBp z that is,

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21 y x z Electromagnetic Waves Electromagnetic waves always travel in the direction of the Poynting vector: Units: W/m 2

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22 Electromagnetic Waves But the direction of the electric and magnetic fields themselves, that is, their polarization, can change y x z Linear polarization

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23 Polarizers 1 2 3 90 o Law of Malus Only a component E p cos of the electric field along the polarization axis can get through

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Electromagnetic Radiation

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25 The Electromagnetic Spectrum

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26 Spectral Response http://landsat.gsfc.nasa.gov/education/compositor Himalyan balsam human beebutterfly

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27 Electromagnetic Radiation An electromagnetic wave carries energy and momentum. The average power per unit area is called the intensity of the wave The momentum per unit time (that is, force) per unit area is called the radiation pressure

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28 Electromagnetic Radiation The radiation pressure, P rad, is given by where the average intensity is given by which can be written in terms of energy density:

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29 The Pressure of Sunshine Solar Luminosity L = 3.8 x 10 26 W Astronomical Unitr= 1.5 x 10 11 m IntensityS= L / 4 r 2 PressureP= S / c

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30 The Pressure of Sunshine IntensityS= L / 4 r 2 = 1370 W/m 2 PressureP= S / c = 4.6 N/m 2

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31 Credit: Michael Carroll, The Planetary Society Interstellar Travel

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32 Summary Maxwell’s Equations 2 nd Unification of forces Electromagnetic waves Universal speed c = 3 x 10 8 m/s Electromagnetic Waves Gamma rays to radio waves Carry energy and momentum Exert pressure

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33 PRS Question 1 The relationship between speed, v, wavelength,, and frequency, f, is 1.v = f/ 2. = f v 3. v = f

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34 PRS Question 2 The 99.9 fm radio signal has a wavelength of about 1.3 km 2.3 m 3. 3 cm

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