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Chapter 24

Chapter 24. Electromagnetic Waves. Unpolarized visible light. X-ray. Radio waves. Polarized visible light. 24.1 Electromagnetic Waves, Introduction. Electromagnetic (EM) waves permeate our environment EM waves can propagate through a vacuum

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Chapter 24

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  1. Chapter 24 Electromagnetic Waves

  2. Unpolarized visible light X-ray Radio waves Polarized visible light

  3. 24.1 Electromagnetic Waves, Introduction • Electromagnetic (EM) waves permeate our environment • EM waves can propagate through a vacuum • Much of the behavior of mechanical wave models is similar for EM waves • Maxwell’s equations form the basis of all electromagnetic phenomena

  4. Conduction Current • A conduction current is carried by charged particles in a wire • The magnetic field associated with this current can be calculated by using Ampère’s Law: • The line integral is over any closed path through which the conduction current passes

  5. Conduction Current, cont. • Ampère’s Law in this form is valid only if the conduction current is continuous in space • In the example, the conduction current passes through only S1 but not S2 • This leads to a contradiction in Ampère’s Law which needs to be resolved

  6. James Clerk Maxwell • 1831 – 1879 • Developed the electromagnetic theory of light • Developed the kinetic theory of gases • Explained the nature of color vision • Explained the nature of Saturn’s rings

  7. Displacement Current • Maxwell proposed the resolution to the previous problem by introducing an additional term called the displacement current • The displacement current is defined as

  8. Displacement current • The electric flux through S2 is EA • S2 is the gray circle • A is the area of the capacitor plates • E is the electric field between the plates • If q is the charge on the plates, then Id = dq/dt • This is equal to the conduction current through S1

  9. Displacement Current • The changing electric field may be considered as equivalent to a current • For example, between the plates of a capacitor • This current can be considered as the continuation of the conduction current in a wire • This term is added to the current term in Ampère’s Law

  10. Ampère-Maxwell Law • The general form of Ampère’s Law is also called the Ampère-Maxwell Law and states: • Magnetic fields are produced by both conduction currents and changing electric fields

  11. 24.2 Maxwell’s Equations, Introduction • In 1865, James Clerk Maxwell provided a mathematical theory that showed a close relationship between all electric and magnetic phenomena • Maxwell’s equations also predicted the existence of electromagnetic waves that propagate through space • Einstein showed these equations are in agreement with the special theory of relativity

  12. Maxwell’s Equations Gauss’ Law (electric flux) Gauss’ Law for magnetism Faraday’s Law of induction Ampère-Maxwell Law The equations are for free space No dielectric or magnetic material is present

  13. Lorentz Force • Once the electric and magnetic fields are known at some point in space, the force of those fields on a particle of charge q can be calculated: • The force is called the Lorentz force

  14. 24.3 Electromagnetic Waves • In empty space, q = 0 and I = 0 • Maxwell predicted the existence of electromagnetic waves • The electromagnetic waves consist of oscillating electric and magnetic fields • The changing fields induce each other which maintains the propagation of the wave • A changing electric field induces a magnetic field • A changing magnetic field induces an electric field

  15. Plane EM Waves • We assume that the vectors for the electric and magnetic fields in an EM wave have a specific space-time behavior that is consistent with Maxwell’s equations • Assume an EM wave that travels in the x direction with the electric field in the y direction and the magnetic field in the z direction

  16. Plane EM Waves, cont • The x-direction is the direction of propagation • Waves in which the electric and magnetic fields are restricted to being parallel to a pair of perpendicular axes are said to be linearly polarized waves • We assume that at any point in space, the magnitudes E and B of the fields depend upon x and t only

  17. Equations of the Linear EM Wave • From Maxwell’s equations applied to empty space, E and B are satisfied by the following equations • These are in the form of a general wave equation, with • Substituting the values for mo and eo gives c = 2.99792 x 108 m/s

  18. Solutions of the EM wave equations • The simplest solution to the partial differential equations is a sinusoidal wave: • E = Emax cos (kx – wt) • B = Bmax cos (kx – wt) • The angular wave number is k = 2 p / l • l is the wavelength • The angular frequency is w = 2 p ƒ • ƒ is the wave frequency

  19. Ratio of E to B • The speed of the electromagnetic wave is • Taking partial derivations also gives

  20. Properties of EM Waves • The solutions of Maxwell’s are wave-like, with both E and B satisfying a wave equation • Electromagnetic waves travel at the speed of light • This comes from the solution of Maxwell’s equations

  21. Properties of EM Waves, 2 • The components of the electric and magnetic fields of plane electromagnetic waves are perpendicular to each other and perpendicular to the direction of propagation • The electromagnetic waves are transverse waves

  22. Properties of EM Waves, 3 • The magnitudes of the fields in empty space are related by the expression • This also comes from the solution of the partial differentials obtained from Maxwell’s Equations • Electromagnetic waves obey the superposition principle

  23. EM Wave Representation • This is a pictorial representation, at one instant, of a sinusoidal, linearly polarized plane wave moving in the x direction • E and B vary sinusoidally with x

  24. Rays • A ray is a line along which the wave travels • All the rays for the type of linearly polarized waves that have been discussed are parallel • The collection of waves is called a plane wave • A surface connecting points of equal phase on all waves, called the wave front, is a geometric plane

  25. Doppler Effect for Light • Light exhibits a Doppler effect • Remember, the Doppler effect is an apparent change in frequency due to the motion of an observer or the source • Since there is no medium required for light waves, only the relative speed, v, between the source and the observer can be identified

  26. Doppler Effect, cont. • The equation also depends on the laws of relativity • v is the relative speed between the source and the observer • c is the speed of light • ƒ’ is the apparent frequency of the light seen by the observer • ƒ is the frequency emitted by the source

  27. Doppler Effect, final • For galaxies receding from the Earth, v is entered as a negative number • Therefore, ƒ’<ƒ and the apparent wavelength, l’, is greater than the actual wavelength • The light is shifted toward the red end of the spectrum • This is what is observed in the red shift

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