The equations so far... Gauss Law for E Fields. Gauss Law for B Fields. B da. inside. d dt. n C 3/28/2018

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1 The equations so far... Gauss Law for E Fields E da S n 1 Q inside Gauss Law for B Fields B da S n C Faraday s Law d E dl dt S B da n Ampere s Law B dl I C 3/8/18 1

2 Ampere s Law B dl I inside _ path No current inside Current inside

3 Mawell s Displacement Current, I d I d de d E da dt dt Putting into changing electric flu just as d dt m C C B dl I I d dt e, this means that a results in a magnetic field, gives rise to an electric field. d 3/8/18 3

4 Mawell s Approach Time varying magnetic field leads to curly electric field. Time varying electric field leads to curly magnetic field? elec elec d dt I elec E nˆ da Q Q Acos A 1 dq 1 I dt delec dt B dl I equivalent current combine with current in Ampere s law I inside _ path 4

5 Mawell s Equations (1865) S S C C E B n n E dl B dl I da da 1 Q :sometimes d dt I S inside S B I n d E t n : Gauss's law da called I S B t Gauss's law for magnetism n da: Faraday's law da: Ampere- Mawell d dt S E n da law 3/8/18 in Systeme International (SI or mks) units 5

Mawell s Equations (Free Space) Note the symmetry of Mawell s Equations in free space, when no charges or currents are present E da B da db d E d E Bd dt dt We can predict the eistence of

6 Mawell s Equations (Free Space) Note the symmetry of Mawell s Equations in free space, when no charges or currents are present E da B da db d E d E Bd dt dt We can predict the eistence of electromagnetic waves. Why? Because the wave equation is contained in these equations. Remember the wave equation. 3/8/18 h 1 h v t h is the variable that is changing in space () and time (t). v is the velocity of the wave. 7

7 3/8/18 8 Review of Waves from Mechanics The one-dimensional wave equation: has a general solution of the form: h(,t) h 1 ( vt) h ( vt) h 1 h v t A solution for waves traveling in the + direction is:

8 Wave Eamples Wave on a String: F d y d d y dt Electromagnetic Wave 3/8/18 v F What is waving?? The Electric & Magnetic Fields!! Rewrite Mawell s equations as equations of the form: The velocity of the wave, v, will be related to and. e.g., sqrt(tension/mass) is wave speed of a guitar string, proportional to frequency of fundamental E z 1 v E t 9

9 Four Step Plane Wave Derivation Step 1 Assume we have a plane wave propagating in z (i.e. E, B not functions of or y) Eample: E E sin(kz t) Step Apply Faraday s Law to infinitesimal loop in -z plane E d d dt E z B y t B y B y E z 1 z Z E z 3/8/18 1

10 Four Step Plane Wave Derivation d B E E d dt z B y t Step 3 Apply Ampere s Law to an infinitesimal loop in the y-z plane: E d E Z Bd z 1 z dt y B y B y Step 4: Use results from steps and 3 to eliminate B y y z 3/8/18 B y tz E t E z B y t B y zt E z E z E t 11

11 Velocity of Electromagnetic Waves We derived the wave equation for E : E E z t The velocity of electromagnetic waves in free space is: 1 v Putting in the measured values for &, we get: v m / s c This value is identical to the measured speed of light! We identify light as an electromagnetic wave. 3/8/18 1

12 Mawell Equations: Electromagnetic Waves Mawell s Equations contain the wave equation The velocity of electromagnetic waves: c = m/s The relationship between E and B in an EM wave Energy in EM waves: the Poynting vector z 3/8/18 y 13

13 Electromagnetic Spectrum ~185: infrared, visible, and ultraviolet light were the only forms of electromagnetic waves known. Visible light (human eye) 3/8/18 16

14 Electromagnetic Spectrum 3/8/18 17

15 Wien s Displacement Law 3/8/18 1 peak Temperature

16 White Light: A Miture of Colors (DEMO) 3/8/18

17 3/8/18 Spectral Lines Energy states of an atom are discrete and so are the energy transitions that cause the emission of a pho (DEMO)

18 How is B related to E? We derived the wave equation for E : We could have derived for B y : By 1 By z c t How are E and B y related in phase and magnitude? Consider the harmonic solution: where kc E 1 E z c t E sin( ) E kz t 3/8/18 5

E & B in Electromagnetic Waves Plane Wave: E sin( ) E kz t B sin( ) y B kz t where: kc E cb z 3/8/18 where y The direction of propagation is given by the cross product

19 E & B in Electromagnetic Waves Plane Wave: E sin( ) E kz t B sin( ) y B kz t where: kc E cb z 3/8/18 where y The direction of propagation is given by the cross product ê, ˆb are the unit vectors in the (E,B) directions. Nothing special about (E,B y ); eg could have (E y, -B ) Note cyclical relation: sˆe ˆbˆ eˆ bˆ sˆ ŝ bˆ sˆe ˆ sˆe ˆbˆ 6

20 Energy in Electromagnetic Waves Electromagnetic waves contain energy. We know the energy density stored in E and B fields: The Intensity of a wave is defined as the average power (P av =u av /t) transmitted per unit area = average energy density times wave velocity: E I c E c For ease in calculation define Z as: 3/8/18 ue 1 E In an EM wave, B = E/c u B 1 B The total energy density in an EM wave = u, where u u u E E B Z c E 1 ub E ue c I E 377 7

21 The Poynting Vector The direction of the propagation of the electromagnetic wave is given by: The intensity for harmonic waves is then given by: 3/8/18 ŝ ê ˆb This energy transport is defined by the Poynting vector S as: E B S S has the direction of propagation of the wave The magnitude of S is directly related to the energy being transported by the wave S EB E c E Z E 377 S E E sin (kz t) 1 Z Z E Z E S E 377 I rms E B Brms 8

22 Characteristics 1. E B. If E E sin( k t), B B sin( k t). E B 3. The Poynting vector S is in the direction of propagation. 4. E and B are both S c m/s (eact) 6. Eo cbo z S 3/8/18 9

Summary of Electromagnetic Radiation combined Faraday s Law and Ampere s Law time varying B-field induces E-field time varying E-field induces B-field E-field

23 Summary of Electromagnetic Radiation combined Faraday s Law and Ampere s Law time varying B-field induces E-field time varying E-field induces B-field E-field and B-field are perpendicular energy density u E B / Poynting Vector describes power flow o S E B / cukˆ units: watts/m B 3 3/8/18 o o E E o o z E t eˆ bˆ sˆ S

Maxwell Equations: Electromagnetic Waves

Maxwell Equations: Electromagnetic Waves Maxwell s Equations contain the wave equation The velocity of electromagnetic waves: c = 2.99792458 x 10 8 m/s The relationship between E and B in an EM wave Energy