Chapter 34 Warmup 20 Ampere’s Law is Not Complete! •Maxwell realized Ampere’s Law is not self-consistent •This isn’t an experimental argument, but a theoretical one •Consider a parallel plate capacitor getting charged by a wire •Consider an Ampere surface between the plates B ds 0 I1 0 •Consider an Ampere surface in front of plates B ds 0 I 2 0 I •But they must give the same answer! •There must be something else that creates B-fields B ds 0 I ? I •Note that for the first surface, there is also an electric field accumulating in capacitor •Maybe electric fields? •Take the time derivative of this formula •Speculate : This replaces I for first surface E Q 0 A 0 E Q I 0 d E dQ I dt dt Ampere’s Law (New Recipe) dE B d s I 0 0 dt B ds 0 I 0 0 dE dt Maxwell’s Equations We now have four formulas that describe how to get electric and magnetic fields from charges and currents E nˆ dA qin 0 •Gauss’s Law S •Gauss’s Law for Magnetism B nˆ dA 0 •Ampere’s Law (final version) S •Faraday’s Law dE •Collectively, these are called B ds 0 I in 0 0 Maxwell’s Equations dt There is also a formula for forces on charges •Called Lorentz Force F q E v B E ds dB dt All of electricity and magnetism is somewhere on this page Warmup 20 Wave solutions •We can solve Maxwell’s Equations (take my word for it) and come up with two “simple” differential equations. E x, y, z , t E0 sin kx t B x, y , z , t B 0 sin kx t 2E 2E 0 0 2 2 x t 2B 2B 0 0 2 2 x t with 0 0 1 2 c •I could have used cosine instead, it makes no difference •I chose arbitrarily to make it move in the x-direction •These are waves where we have . 2 f f c k 2 http://www.ariel.ac.il/sites /cezar/ariel/ANIMATION S/physics3.html Wave Equations summarized: E x, y, z , t E0 sin kx t •Waves look like: B x, y , z , t B 0 sin kx t ck •Related by: •Two independent solutions to these equations: E0 cB0 B0 E0 E0 B0 •Note E B is in direction of motion E y 0 cBz 0 or Ez 0 cE y 0 •Note that E, B, and direction of travel are all mutually perpendicular •The two solutions are called polarizations •We describe polarization by telling which way E-field points Understanding Directions for Waves E0 cB0 •The wave can go in any direction you want •The electric field must be perpendicular to the wave direction •The magnetic field is perpendicular to both of them •Recall: E B is in direction of motion A wave has an electric field given by E = j E0 sin(kz – t). What does the magnetic field look like? A) B = i (E0/c) sin(kz - t) B) B = k (E0/c) sin(kz - t) C) B = - i (E0/c) sin(kz - t) D) B = - k (E0/c) sin(kz - t) •The magnitude of the wave is B0 = E0 / c •The wave is traveling in the z-direction, because of sin(kz - t). •The wave must be perpendicular to the E-field, so perpendicular to j •The wave must be perpendicular to direction of motion, to k •It must be in either +i direction or –i direction •If in +i direction, then E B would be in direction j i = - k, wrong •So it had better be in the –i direction CT – 2 A planar electromagnetic wave is propagating through space. Its electric field vector is given by E = Eo cos(kz – wt) î . Its magnetic field vector is A. B = Bo cos(kz – wt) ˆj B. B = Bo cos(ky – wt) k̂ C. B = Bo cos(ky – wt) î D. B = Bo cos(kz – wt) k̂ E. None of the above The meaning of c: ck E x, y, z , t E0 sin kx t •Waves traveling at constant speed B x, y , z , t B 0 sin kx t •Keep track of where they vanish kx t 0 x k t ct •c is the velocity of these waves c 1 0 0 2.99792458 108 m/s c 3.00 108 m/s •This is the speed of light •Light is electromagnetic waves! •But there are also many other types of EM waves •The constant c is one of the most important fundamental constants of the universe Wavelength and wave number •The quantity k is called the wave number •The wave repeats in time •It also repeats in space f 1 T k 2 E E0 sin kx t B B 0 sin kx t 2 f ck •EM waves most commonly described in terms of frequency or wavelength c 2 f k 2 cf •Some of these equations must be modified when inside a material Ex- (Serway 34-19) In SI units, the electric field in an electromagnetic wave is described by Ey = 100 sin(1.00 x 107x -t). (a) Calculate the amplitude of the corresponding magnetic field. (b) Find the wavelength , (c) Find the frequency f. Also find an expression for the magnetic field. Solve on Board Energy and the Poynting Vector •Let’s find the energy density in the wave E0 cB0 u E 12 0 E 2 12 0E02 sin 2 kx t 12 0c2 B02 sin 2 kx t 1 0 B02 sin 2 kx t 2 B 2 0 0 2 0 u sin kx t 2 2 B0 B 0 uB sin 2 kx t 2 0 2 0 1 •Now let’s define the Poynting vector: S E B 1 0 2 S E0 B0 sin kx t cu 0 •It is energy density times the speed at which the wave is moving •It points in the direction energy is moving •It represents the flow of energy in a particular direction •Units: J m W S uc m3 s m2 Intensity and the Poynting vector •The time-averaged Poynting vector is called the Intensity •Power per unit area 2 S c u cB0 0 sin 2 S cB02 20 kx t In Richard Williams’ lab, a laser can (briefly) produce 50 GW of power and be focused onto a region 1 m2 in area. How big are the electric and magnetic fields? P 5.0 1010 W 22 2 S 5.0 10 W/m 2 6 A 10 m 7 22 2 2 4 10 T m/A 5.0 10 W/m 2 0 S 2 B0 3 108 m/s c 4.2 108 T 2 12 E cB 0 0 E 6.1 10 V/m B 20 kT 0 0 The Electromagnetic Spectrum •Different types of waves are classified by their frequency (or wavelength) Increasing f Increasing cf Radio Waves Microwaves Infrared Visible Ultraviolet X-rays Gamma Rays •Boundaries are arbitrary and overlap •Visible is 380-740 nm Red Which of the Vermillion following waves Orange hasSaffron the highest Yellow speed in vacuum? Chartreuse A) Green Infrared B) Orange Turquoise Blue C) Green Indigo Violet D) Blue E) It’s a tie F) Not enough info Sources of EM waves + •A charge at rest produces no EM waves •There’s no magnetic field •A charge moving at uniform velocity produces no EM waves •Obvious if you were moving with the charge •An accelerating charge produces electromagnetic waves •Consider a current that changes suddenly •Current stops – magnetic field diminishes •Changing B-field produces E-field •Changing E-field produces B-field •You have a wave –