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Lecture 21-1 Resonance For given peak, R, L, and C, the current amplitude Ipeak will be at the maximum when the impedance Z is at the minimum. peak I peak R X L X C 2 2 Z X L XC res L 1 resC i.e., load purely resistive This is called resonance. Resonance angular frequency: 1 res LC , Z R, and I peak peak R ε and I in phase Lecture 21-2 Transformer • AC voltage can be stepped up or down by using a transformer. • AC current in the primary coil creates a time-varying magnetic flux through the secondary coil via the iron core. This induces EMF in the secondary circuit. Ideal transformer (no losses and magnetic flux per turn is the same on primary and secondary). (With no load) V1 L 0 d B V1 V2 turn dt N1 N 2 N1 N 2 V1 V2 N1 N 2 V1 V2 step-up step-down With resistive load R in secondary, current I2 flows in secondary by the induced EMF. This then induces opposing EMF back in the primary. The latter EMF must somehow be exactly cancelled because is a defined voltage source. This occurs by another current I1 which is induced on the primary side due to I2. Lecture 21-3 Maxwell’s Equations (so far) Gauss’s law Gauss’ law for magnetism S E dA S Qinside 0 B dA0 Faraday’s law dB C E d l dt Ampere’s law* C B d l 0 I Lecture 21-4 Parallel-Plate Capacitor Revisited -Q S B d l 0 I S Q E E 0 dQ dV 0 A dV C dt dt d dt dE dE 0 A 0 dt dt I dE Id 0 dt will work. Lecture 21-5 Displacement Current James Clerk Maxwell proposed that a changing electric field induces a magnetic field, in analogy to Faraday’s law: A changing magnetic field induces an electric field. Ampere’s law is revised to become Ampere-Maxwell law dE C B d l 0 ( I I d ) 0 I 0 0 dt where dE Id 0 dt is the displacement current. Lecture 21-6 Maxwell’s Equations S E dA Qinside 0 dB C E d l dt S B dA0 dE C B d l 0 I 0 0 dt Basis for electromagnetic waves! Lecture 21-7 Electromagnetic Waves From Faraday’s Law dB C E dl dt E y Bz x t B Bm sin(kx t ) z E Em sin(kx t ) y Em Bm k EB k c Lecture 21-8 Electromagnetic Waves From Ampère’s Law dE C B dl 0 0 dt E y Bz 0 0 x t E Em sin(kx t ) y B Bm sin(kx t ) z Em k / c Bm 0 0 EB k Lecture 21-9 Electromagnetic Wave Propagation in Free Space So, again we have a traveling electromagnetic wave Em c Bm k c Em 1 Bm 0 0c 0 4 107 (T m / A) 1 0 0 speed of light in vacuum 0 8.85 1012 (C 2 / N m2 ) B 1 E 2 x c t E B x t Ampere’s Law Faraday’s Law 2 B 1 2 B Wave Equation 2 2 2 x c t c 3.00 108 (m / s) Speed of light in vacuum is currently defined rather than measured (thus defining meter and also the vacuum permittivity). Lecture 21-10 Plane Electromagnetic Waves 2B 1 2B 2 2 2 x c t 2E 1 2E 2 2 2 x c t where B Bm sin(kx t ) z E Em sin(kx t ) y EB k • Transverse wave • Plane wave (points of given phase form a plane) x • Linearly polarized (fixed plane contains E) Lecture 21-11 Non-scored Test Quiz Electromagnetic wave travel in space where E is electric field, B is magnetic field. Which of the following diagram is true? z z (a). (b). E y x travel direction travel direction B E y x B z travel direction z (c). (d). E x B E travel direction y B x y Lecture 21-12 Energy Density of Electromagnetic Waves • Electromagnetic waves contain energy. We know already expressions for the energy density stored in E and B fields: EM wave 2 1B 1 2 uE 0 E , uB 2 0 2 E2 1 2 uB E uE 0 2 20c 2 Bm Em / c B E/c • So Total energy density is EB 0 u uE uB 0 E EB 0 0c 0 B2 2 u 0 E 2 0E 2 rms B2 0 2 Brms 0 EB Erms Brms 0 c 0 c Lecture 21-13 Energy Propagation in Electromagnetic Waves • Energy flux density = Energy transmitted through unit time per unit area • Intensity I = Average energy flux density (W/m2) P u c A Erms Brms 1 EB 0 Define Poynting vector 0 S 1 0 EB Direction is that of wave propagation average magnitude is the intensity S I c 0 E 2 1 1 Bm2 Em Bm 2 c 0 Em c 2 2 0 2 0 Lecture 21-14 Radiation Pressure Electromagnetic waves carry momentum as well as energy. In terms of total energy of a wave U, the momentum is U/c. During a time interval t , the energy flux through area A is U =IA t . If radiation is totally absorbed: p U / c IA / c t p IA F t c momentum imparted pr F / A I / c B /(20 ) 2 m radiation pressure EXERTED If radiation is totally reflected: p 2U / c 2 IA / c t 2 IA F , pr 2 I / c c Lecture 21-15 Maxwell’s Rainbow Light is an Electromagnetic Wave f c Lecture 21-16 Physics 241 –Quiz 18b – March 27, 2008 An electromagnetic wave is traveling through a particular point in space where the direction of the electric field is along the +z direction and that of the magnetic field is along +y direction at a certain instant in time. Which direction is this wave traveling in? a) +x b) x c) y d) z e) None of the above Lecture 21-17 Physics 241 –Quiz 18c – March 27, 2008 An electromagnetic wave is traveling in +y direction and the magnetic field at a particular point on the y-axis points in the +z direction at a certain instant in time. At this same point and instant, what is the direction of the electric field? a) z b) x c) y d) +x e) None of the above