Maxwell s equations and EM waves. From previous Lecture Time dependent fields and Faraday s Law

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1 Maxwell s equations and EM waves This Lecture More on Motional EMF and Faraday s law Displacement currents Maxwell s equations EM Waves From previous Lecture Time dependent fields and Faraday s Law 1

2 Radar Remote Sensing of the Atmosphere Stan Briczinski April 4, 2008

3 Calculation of Electric/Magnetic Field for a moving charge Coulomb's Law: de = 1 4πε 0 dq r 2 u r + Biot-Savart db = µ 0 4π I ds u r r 2 I Now charge moves inside page 3

4 Gauss law in magnetostatics Net magnetic flux through any closed surface is always zero (the number of lines that exit and enter the closed surface is equal) B da = 0 No magnetic charge, so right-hand side=0 for B-field Basic magnetic element is the dipole Compare to Gauss law for E-field E da = Q enclosed ε o 4

5 Lorentz force in E and B-fields If a charge moves in the presence of E-field and B-field r r F = qe + qv r r B velocity selector If we switch on an E-field parallel to B it will accelerate q and loops become more stretched as q gains speed 5

6 Summary on static E- and B-fields Magnetostatics Gauss law B da = 0 Surface Ampere s law B ds Line = µ o I Electrostatics E da Surface = Q enclosed ε o E ds =? 0 Line Ampere s law true for static (no time dependence) electric fields. What happens to the Ampere s law if fields depend on time? 6

7 Time-dependent B-field E ds = 0 becomes E ds E V Faraday s law = 0 dφ B dt Not only charges produce E-field a changing B-field also produces an E-field. This is not like a Coulomb field produced by a charge starting or stopping in the charge but it is a circular E-field 7

8 Remember last lecture Faraday s Law Lenz s law ε = E ds = d dt Φ B = d dt B da E ds Integral over closed path E is not conservative!!! Magnetic flux through surface bounded by path This circular E-field is able to move charges around the loop around the changing B-field 8

9 Quick Quiz A rectangular loop of wire is pulled at a constant velocity from a region with B = 0 into a region of a uniform B-field. The induced current; A) will be zero B) will be some constant value that is not zero C) will increase linearly with time What happens when the loop exits the B-field region? L Iind v I = ε R = dφ B dt = BL dx dt = BLv Bind x 9

10 Faraday s and Lenz s law B flux increases when magnet moves since B from N to S Bind opposite to B S N B Bind v B flux decreases when magnet moves, Bind parallel to B and I changes direction S v N Bind B Magnetic flux change: the magnitude of B changes in time 10

11 Other ways to change B-flux 2) the area crossed by B lines changes with t (motional emf) 3) The angle θ between B and normal to loop changes with t Φ B = BAcosθ 11

12 Quick Quiz: motional emf A conducting bar moves with velocity v. Which statement is true? A) + accumulates at the top, and - at the bottom B) no complete circuit, therefore, no charge accumulation C) - accumulates at the top, and + at the bottom + - v L + - v F B =-e vxb THERE IS AN ELECTRIC FIELD ACROSS THE BAR AND A POTENTIAL DIFFERENCE AT EQUILIBRIUM qe = qvb or E = vb Potential difference = v B L 12

13 Induced current in a moving rod Equivalent circuit Moving bar acts as a battery: R resistance of circuit F B = I r l B r force due to induced current on bar (drag force opposing to bar motion)! 13

14 Ampere s laws with time-dep fields Faraday s law E ds = 0 becomes E ds = 0 dφ B dt Not only charges produce E-field, a changing B-field produces a non conservative E-field Ampere-Maxwell s law B ds = µ o I becomes B ds E V dφ = µ o I + µ o ε? E o dt a changing E-field produces a B-field 14

15 Displacement current Ampere s law says we can consider any surface bounded by close line C BUT current through S1 is I 0 and current through S2 I=0 Is there something wrong? There is an electric flux through S2 Φ E = E da = E da = q S2 ε 0 S 1 +S 2 A is the area of the capacitor plates C I conduction current in wire Displacement current = conduction current Id is given by the variation of E 15

16 Ampere s - Maxwell Law Tot current is conduction + displacement +µ 0 I d B ds = µ o I becomes B ds = µ o I + µ o ε o dφ E dt Magnetic fields are produced both by conduction currents and by time-varying electric fields 16

17 The 4 Maxwell s Equations charged particles create an electric field E da = q (Gauss' Law) B da = 0 S ε S 0 There are no magnetic monopoles L E ds = dφ B dt (Faraday - Henry) B ds = µ L 0 I + µ 0 ε 0 dφ E dt (Ampere - Maxwell law) An E-field can be created by a changing B-field Consequence: induced current Lorentz force F = q(e + v B) Currents create a B-field A changing E-field can create a B-field 17

18 James Clerk Maxwell and the theory of electromagnetism Electricity and magnetism are deeply related 1865 Maxwell: mathematical theory showing relationship between electric and magnetic phenomena Maxwell s equations predict existence of electromagnetic waves propagating through space at velocity of light = x 10 8 m/s Permittivity of free space: Permeability of free space: 18

19 Reminders: classification of waves A mechanical wave is a disturbance created by a vibrating object that travels through a medium from one location to another Longitudinal wave: medium particles move in direction parallel to direction of propagation of wave (eg. sound waves) Sound wave: sinusoidal variation of pressure in air 19

20 Reminders: waves Transverse wave: medium particles move in direction perpendicular to direction of wave (waves on a pond, EM waves) Velocity of waves: λ=vt v = λ/t=λf 20

21 Reminders on waves Traveling waves on a string along x obey the wave equation: 2 y(x,t) x 2 = 1 v 2 2 y(x,t) t 2 y=wave function General solution : y(x,t) = f1(x-vt) + f2(x+vt) y = displacement λ pulse traveling along +x pulse traveling along -x Traveling wave: superposition of sinusoidal waves (produced by a source that oscillates with simple harmonic motion): y(x,t) = A sin(kx-ωt) y(x,t) = sin(kx+ ωt) A = amplitude k = 2π/λ = wave number λ = wavelength v f = frequency T = 1/f = period ω = 2πf=2π/T angular frequency 21

22 Fields exist even with q=0 and I=0! E da = 0 (Gauss' Law) B da = 0 S S L E ds = dφ B dt (Faraday - Henry) B ds = L dφ µ 0 ε E 0 dt (Ampere - Maxwell law) From these equations we get EM wave equations traveling in vacuum! λ E x B direction of c Transverse wave composed of E and B fields propagating in empty space with velocity c emerges from Maxwell equations! 22

23 Solutions of Maxwell s equations λ The simplest solution to partial differential equations is sinusoidal wave propagating along x: E = E max cos (kx ωt) B = B max cos (kx ωt) E B = 0 E and B are perpendicular at all times The speed of the electromagnetic wave is 23

24 Hertz confirms Maxwell s predictions Heinrich Hertz was the first to generate and detect electromagnetic waves in a laboratory setting The most important discoveries were in 1887 He also showed other wave aspects of light 24

25 Hertz s Experiment (1887) T R Transmitter T: two spherical electrodes separated by narrow gap charged until air gap ionized=>sparks The discharge has oscillatory behavior of frequency f ~ 4 x 10 7 Hz This oscillating dipole emits E and B plane waves When resonance frequency of T and R match sparks also in R = receiver Hertz s hypothesis: the energy transmitted from T to R is carried by waves measured λ by having standing waves using a metal screen: nodes at distance λ/2 knowing f he measured c!) Radiation has wave properties: interference, diffraction, reflection, refraction and polarization 25

26 Quick Quiz on EM Waves Shown below is the E-field of an EM wave broadcast at 30 MHz and traveling to the right. What is the direction of the magnetic field during the first λ/2? 1) Into the page 2) out of the page E x λ/ What is the wave length? 1) 10 m 2) 5 m = m /s /s =10m

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