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Understanding Maxwell's Equations and Electromagnetic Waves

Learn about Maxwell's equations and how they explain the existence of electromagnetic waves, including polarization, energy transport, and momentum. Explore the connection between electric and magnetic fields and the speed of light. Understand the effects of polarization by scattering and reflection. Discover the wave equation for electromagnetic waves and the concept of radiation pressure.

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Understanding Maxwell's Equations and Electromagnetic Waves

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  1. 30 Outline • Maxwell’s Equations and the Displacement Current • Electromagnetic Waves • Polarization

  2. Maxwell’s Equations • Gauss’ Law: E & B • Faraday’s Law • Ampere’s Law

  3. displacement current • ‘explains’ existence of B around E

  4. EM waves • accelerating charges produce ‘waves’ of E and B • can be ‘pulse’ or ‘harmonic wave’

  5. dipole radiation

  6. EM waves • transverse: E perpendicular to B • E and B are in phase • speed: c = fl = 3  108 m/s

  7. Electric Dipole Radiation Example: I(r = 1.0m, angle = 90) is 12 W/m2. I at 2.0m and angle of 30 degrees is:

  8. antennas • can respond to E or B

  9. Light Light is an electromagnetic wave c = lf ≈ 3  108 m/s As light waves travel through space they: • transport energy and/or information • transport momentum

  10. EM Waves • carry energy and momentum, shared equally between the electric and magnetic fields.

  11. Energy and Momentum in EM Waves • Intensity: energy/area/time [watts/sq.meter]

  12. Example 50W Bulb • Assume that 5.00% of the electrical power consumed by the bulb is converted to light. The average intensity at a distance of r = 1.00m: • The rms value of E:

  13. Polarization • Unpolarized light is the superposition of many waves with random direction of E. • Linearly Polarized light has only one direction of E.

  14. Polarizing Filters • Polarizing material only allows the passage of only one direction of E • Malus’ Law:

  15. Two Filters (incident light unpolarized) • 1st reduces intensity by 1/2. • 2nd reduces according to Malus’ Law

  16. Polarization by Scattering and Reflection • Light scattered at 90 degrees is 100% polarized.

  17. Summary • displacement current added to Ampere’s Law: completes Maxwell Eqs., which explain ‘light’ properties (transverse EM wave with speed c) • visible light small segment of spectrum • energy density and pressure • polarization by reflection/scattering • Malus’ Law

  18. Omit 30-4The Wave Equation for Electromagnetic Waves End

  19. Momentum • momentum = U/c The total energy received in the time by an area A The momentum received The average force Radiation pressure

  20. Example (cont.) Part (b) 2. Use to find 3. Use to find

  21. Example: Consider a laser that puts out an average power of P=1.0 milliwatt in a beam having a diameter of 1.0 mm. What is the peak amplitude of the electric field? The area of the laser beam is The electromagnetic flux S is Recall so that Substitution of the numerical values yields and thus

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