* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Chapter 3 Electromagnetic Theory, Photons, and Light
Superconductivity wikipedia , lookup
Electromagnetic mass wikipedia , lookup
Casimir effect wikipedia , lookup
Coherence (physics) wikipedia , lookup
Nordström's theory of gravitation wikipedia , lookup
Speed of gravity wikipedia , lookup
Diffraction wikipedia , lookup
Field (physics) wikipedia , lookup
Introduction to gauge theory wikipedia , lookup
Lorentz force wikipedia , lookup
Maxwell's equations wikipedia , lookup
Circular dichroism wikipedia , lookup
Thomas Young (scientist) wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Time in physics wikipedia , lookup
Wave packet wikipedia , lookup
Electromagnetism wikipedia , lookup
Photon polarization wikipedia , lookup
Wave–particle duality wikipedia , lookup
Electromagnetic radiation wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Lecture 5 Chapter 3 Electromagnetic Theory, Photons, and Light Maxwell equations Gauss’s B dS 0 Gauss’s 1 E dS q S S In matter d Faraday’s CE dl AB dS dt E Ampère dS B dl J A Maxwell’s C t + fields are defined through interaction with charges Lorentz force: F qE qv B Maxwell equations: free space, no charges Current J and charge are zero Integral form of Maxwell equations in free space: no magnetic ‘charges’ SB dS 0 no electric charges SE dS 0 dB changing magnetic field E dl dS C A creates changing electric dt field E changing electric field CB dl 0 0 A t dS creates changing magnetic field There is remarkable symmetry between electric and magnetic fields! Maxwell equations: differential form (free space) E 0 B 0 B E t E B 0 0 t ˆ ˆ ˆ i j k Notation: x y z 2 2 2 2 Laplacian: 2 2 2 x y z E x E y E z E div ( E ) 0 y x z E z E y E x E z E y E x E iˆ kˆ ˆj z z x x y y E curl (E ) E E B z y x y z t B y E x E z t x z E y E x B z t y x Electromagnetic waves (free space) B E t E B 0 0 t Changing E field creates B field Changing B field creates E field Is it possible to create self-sustaining EM field? Can manipulate mathematically into: 2 2 E B 2 2 E 0 0 2 B 0 0 2 t t Electromagnetic waves E 2 E 0 0 2 t ˆ ˆ j k iˆ x x x 2 2 2 2 2 2 2 z x y B 2 B 0 0 2 t 2 2 2 2 1 2 Resembles wave equation: 2 2 2 2 2 z x y v t 2 2 2 Ex 2 Ex 2 Ex 2 Ex 0 0 2 2 2 2 x y z t 2 Bx 2 Bx 2 Bx 2 Bx 0 0 2 2 2 2 x y z t 2Ey 2Ey 2 By t x x 2 2Ey y 2 2Ey z 2 0 0 2 2 Ez 2 Ez 2 Ez 2 Ez 0 0 2 2 2 2 x y z t 2 2 By y 2 2 By z 2 0 0 2 By t 2 2 Bz 2 B z 2 Bz 2 B z 0 0 2 2 2 2 x y z t Each component of the EM field obeys the scalar wave equation, 1 provided that v 0 0 Light - electromagnetic wave? Maxwell in ~1865 found that EM wave must move at speed v 1 0 0 At that time permittivity 0 and permeability 0 were known from electric/magnetic force measurements and Maxwell calculated v 1 0 0 310,740 km/s Speed of light was also measured by Fizeau in 1849: 315,300 km/s Maxwell wrote: This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws. celer (lat. - fast) Exact value of speed of light: c = 2.997 924 58 × 108 m/s Electromagnetic wave Assume: reference frame is chosen so that E=(Ex,0,0) longitudinal wave, propagates along x E 0 B 0 E x E y E z 0 z x y B E t E B 0 0 t E x 0 x Ex does not vary with x This cannot be a wave! Conclusion: it must be transverse wave, i.e. Ex=0. Similarly Bx=0. Since E is perpendicular to x, we must specify its direction as a function of time Direction of vector E in EM wave is called polarization Simple case: polarization is fixed, i.e. direction of E does not change Polarized electromagnetic wave We are free to chose y-axis so that E field propagating along x is polarized along y: (0, Ey ,0). E y Bz E 0 Bx Ez E y t y z x t B 0 By Ex Ez Also: Bx=By=const (=0) B t z x E E y Ex Bz t x y t E B 0 0 t E-field of wave has only y component B-field of wave has only z component (for polarized wave propagating along x) In free space, the plane EM wave is transverse Harmonic polarized electromagnetic wave Harmonic functions are solution for wave equation: E y x, t E0 y cos t x / c polarized along y axis propagates along x axis Find B: E y Bz x t Bz E y x dt 1 Bz x, t E0 y cos t x / c c E y cBz This is true for any wave: - amplitude ratio is c - E and B are in-phase Harmonic polarized electromagnetic wave Electromagnetic waves * direction of propagation is in the direction of cross-product: EB * EM field does not ‘move’ in space, only disturbance does. Changing E field creates changing B field and vice versa Energy of EM wave It was shown (in Phys 272) that field energy densities are: 0 2 u 1 B2 uE E B 2 0 2 Since E=cB and c=(00)-1/2: uE uB - the energy in EM wave is shared equally between electric and magnetic fields Total energy: u uE uB 0 E 2 1 0 B 2 (W/m2) The Poynting vector EM field contains energy that propagates through space at speed c Energy transported through area A in time t: uAct Energy S transported by a wave through unit area in unit time: E c2 1 1 uAct 2 0 EB EB S uc c 0 E c 0 E cB 0 0 0 At John Henry Poynting (1852-1914) The Poynting vector: 1 S EB 0 power flow per unit area for a wave, direction of propagation is direction of S. (units: W/m2) The Poynting vector: polarized harmonic wave 1 S EB Polarized EM wave: E E0 cos k r t B B0 cos k r t Poynting vector: 1 2 S E0 B0 cos k r t 0 0 This is instantaneous value: S is oscillating Light field oscillates at ~10 15 Hz most detectors will see average value of S. Irradiance (used to be called Intensity) Average value for periodic function: need to average one period only. It can be shown that average of cos2 is: And average power flow per unit time: Irradiance: c 0 2 I S T E0 2 1 2 S E0 B0 cos k r t 0 cos 2 t 1 2 T 1 c 0 2 S T E0 B0 E0 2 0 2 Alternative eq-ns: c 2 I c 0 E B2 T 0 For linear isotropic dielectric: T I v E 2 T Irradiance is proportional to the square of the amplitude of the E field Usually mostly E-field component interacts with matter, and we will refer to E as optical field and use energy eq-ns with E Optical power radiant flux total power falling on some area (Watts) Spherical wave: inverse square law Spherical waves are produced by point sources. As you move away from the source light intensity drops Spherical wave eq-n: A r , t cosk r vt r E0 B0 E cos k r t B cos k r t r r 1 E0 B0 2 cos k r t S 0 r r I S T c 0 1 2 E0 2 2 r Inverse square law: the irradiance from a point source drops as 1/r2 Radiation pressure Using classical EM theory Maxwell showed that radiation pressure equals the energy density of the EM waves: P u 0 2 E 2 1 2 0 B 2 S t P t c S uc This is the instantaneous pressure that would be exerted on a perfectly absorbing surface by a normally incident beam Average pressure: P t T S t T c I c (N/m2) * for reflecting surface pressure doubles Radiation pressure application Star wars episode 2 NASA to Launch World's Largest Solar Sail in November 2014: Sunjammer http://www.sunjammermission.com http://www.livescience.com/32593how-do-solar-sails-work-.html Example problem A laser pointer emits light at 630 nm in xy plane at =450 to axis x (counter clock-wise). The light is polarized along axis z , beam cross-section is A=1 mm2 and its power is P=1 mW. 1. Write an equation of E and B components of this EM wave for the region of the beam. z E E E0 cos k r t y B Find : 2 2c Find k: k 2 iˆ cos ˆj sin Find E0: P c 0 2 x I E0 Irradiance: A 2 E0 2 P Ac 0 E0 2 P Ac 0 k̂ Electric field: E 2c 2P 2 ˆ ˆ i cos j sin r t k̂ cos Ac 0