29. Maxwell’s Equations

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

3 Laws of Electric & Magnetic Fields

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, (1865). This is one of the greatest scientific papers ever written.

5 Maxwell’s Equations

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

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

The 2 nd Unification of Forces

9  0 is the magnetic constant  0 is the electric constant

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

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)

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

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

Electromagnetic Waves

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

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

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:

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

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

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

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

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

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

Electromagnetic Radiation

25 The Electromagnetic Spectrum

26 Spectral Response Himalyan balsam human beebutterfly

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

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:

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

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

31 Credit: Michael Carroll, The Planetary Society Interstellar Travel

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

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

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