1 Chapter 32

2 Maxwell’s Equations The electric field spreads into space proportional to the amount of static charge and how closely you space the static charges Magnetic field lines are closed loops and always return to the source creating them An electric field, resembling a magnetic field in shape, can be created by a time-varying magnetic field. There are two ways to produce a magnetic field: 1) by a current and 2) by a time-varying electric field.

3 Maxwell’s Equations for Vacuum In vacuum, there is no charge so r enc =0 Since no charge, no currents enclosed so J=0 Note the symmetry of the equations i.e. they look practically the same!!

4 Without proof

5 Separating E from B

6 The Wave Equation This equation is called a wave equation. In order to simplify the math, let’s just work with 1-dimension i.e. in the x- direction

7 A solution is a sine function

8 Using the wavelength and wave number fl is the speed of the wave, which we will call c

9 An important revelation

10 However, I could have solved for B

11 Very similar functions So the solution for E and B are mathematically similar Now, let’s assign a direction for E in the y- direction

12 Using the curl

13 Implications The wave is called transverse; both E and B are perpendicular to the propagation of the wave. The direction of propagation is in the direction of E x B. E=cB The wave travels in vacuum with a definite and unchanging speed What is the wave propagating through?

14 “Common Sense” on Waves Ocean waves propagate in water Sound waves propagate in air Mechanical waves propagate through material where they are transmitting Ergo, the 19 th Century physicists thought that EM waves propagate through the “ether”. Ether surrounds us and we move through it without any drag.

15 Michelson-Morley Experiment In 1887, Michelson and Morley invented an experiment to measure the speed of light in the direction of Earth’s motion and in the direction against Earth’s motion If there is ether, then there should be a slight difference in the speed of light. Michelson-Morley found NO evidence of any difference of the speed of light. Why?

16 The New Physics Actually, if they trusted their equations, they would have seen that there is no need to have a medium

17 A completely different direction Einstein thought about the results of the MM experiment. He assumed that there was no mistake and the c is always constant The Postulates of Special Relativity 1. The laws of physics are the same in every inertial frame of reference 2. The speed of light in vacuum is the same in all inertial frames of reference and is independent of the motion of the source This is the beginning of the new physics of the 20 th century From here, we can get E=mc 2 and from there, quantum mechanics

18 Waves in Matter Recall  =k  0  =  m   c=1/sqrt(    0 ) Let v=speed of light in a material v=1/sqrt(  ) < c v=c/sqrt(k*  m ) Index of refraction, n n=c/v=sqrt(k*  m )

19 Radiation Pattern The Poynting Vector, S,describes the energy flow per unit area and per unit time through a cross-sectional area perpendicular to propagation direction S=(E x B)/  0 The “intensity” of the EM wave in vacuum is defined as I=S av =(E 0 B 0 )/2  0 = ½  0 cE 0 2

20 EM Spectrum