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Physics 272 April 10 Spring 2014 http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html Prof. Philip von Doetinchem [email protected] Phys272 - Spring 14 - von Doetinchem - 258 Summary Phys272 - Spring 14 - von Doetinchem - 259 Resonance in alternating-current circuits ● ● ● It is important to understand how LRC circuits depend on the angular frequency Example: radio signal of a certain frequency produces greatest signal when the circuit is tuned to this frequency → circuit is in resonance Connect AC source with constant voltage and variable frequency to LRC series circuit – Current has same frequency – Impedance depends on frequency Phys272 - Spring 14 - von Doetinchem - 264 Resonance in alternating-current circuits minimum of impedance ● A minimum in impedance exists where X L and XC cancel ● Current becomes maximum at this frequency I=V/Z Phys272 - Spring 14 - von Doetinchem - 265 Circuit behavior at resonance ● Position of minimum impedance is called resonance frequency – ● ● ● Capacitive and inductive reactance cancel: This is the same frequency of an ideal LC circuit Inductive and capacitive voltages have a 180deg phase angle → cancel at all times at resonance frequency Voltage across resistor is equal to source voltage at resonance (circuit behaves like there are no inductors or capacitors) Phys272 - Spring 14 - von Doetinchem - 266 Tailoring an AC circuit ● ● ● Resonance frequency can be changed by changing L and/or C Tuning knob in old radios was moving capacitor plates → change in capacitance Source: http://commons.wikimedia.org/wiki/File:Autoradio_De_Wald.jpg Modern approach: change L by moving ferrite core Phys272 - Spring 14 - von Doetinchem - 267 Tailoring an AC circuit https://www.youtube.com/watch?v=ZYgFuUl9_Vs Phys272 - Spring 14 - von Doetinchem - 268 ● ● ● ● Sharp increase at resonance frequency current Tailoring an ac circuit R Value of resistance determines how sharp the change is Important for discriminating between different radio stations If the peak is too sharp → information will be lost 3xR 9xR frequency Phys272 - Spring 14 - von Doetinchem - 269 Tuning a radio ● Resonance frequency: ● Reactances: Phys272 - Spring 14 - von Doetinchem - 270 Tuning a radio ● RMS current: ● Voltages: VL, VC are both significantly larger than V R → they cancel at resonance (180deg phase angle) Phys272 - Spring 14 - von Doetinchem - 271 Transformers ● Advantage of AC over DC electric power distribution: – ● ● Stepping voltage levels up and down is easy Long distance transmission: – high voltages (500kV) → lower i2R losses at the same energy throughput – Thinner cables for high voltages Safety requires that the normal user operates at much lower voltages Phys272 - Spring 14 - von Doetinchem - 274 How transformers work ● ● ● AC source causes an alternating current in the primary coil (the one that is powered) Alternating magnetic flux in core (made of material with high permeability) → induced emf in primary coil Same alternating flux goes through secondary coil → also induces emf at same frequency as primary coil Phys272 - Spring 14 - von Doetinchem - 275 How transformers work ● Magnetic flux through both coils is the same: ● Induced emf is equal to terminal voltage if windings have zero resistance Phys272 - Spring 14 - von Doetinchem - 276 How transformers work ● ● Choice of windings determines the voltage output of the secondary coil Wall charger transforms 110V AC to, e.g., 5V DC for USB Phys272 - Spring 14 - von Doetinchem - 277 Energy considerations for transformers ● ● ● Power delivered to the primary coil is the same that is taken out by the secondary to power a device Not only voltages are transformed, but also the impedance Real transformers have losses due to – non-zero resistance (transformer gets warm) – Hysteresis in the core Phys272 - Spring 14 - von Doetinchem - 278 Energy considerations for transformers ● ● Real transformers have losses due to – Magnetic field is constantly changing → eddy currents build up → energy wasted – Use laminated cores to narrow path for eddy-currents → smaller radius → lower opposing magnetic field – Magnetic flux in each path is much smaller Transformers typically reach efficiencies of greater than 90% Phys272 - Spring 14 - von Doetinchem - 279 Review ● Root-mean-squares of sinusoidal voltages and currents for ● Resistance and reactance: ● Impedance: ● Power in an AC circuit: Phys272 - Spring 14 - von Doetinchem - 282 Review ● Resonance in an AC circuit ● Transformers: Phys272 - Spring 14 - von Doetinchem - 283 Electromagnetic waves ● What is light? → electromagnetic wave → electromagnetism is needed (not the complete story → QFT) ● Time varying magnetic field creates electric field Time varying electric field creates magnetic field → sustain each other and create an electromagnetic wave that propagates through space Phys272 - Spring 14 - von Doetinchem - 284 Electromagnetic waves Source: http://en.wikipedia.org/wiki/Light ● Electromagnetic waves carry energy and momentum ● Electric and magnetic fields in sinusoidal waves have defined frequency ● ● Infrared, visible light, UV, X-ray, -ray, etc. all follow the same principle, but at different wavelength Electromagnetic waves do not require medium (like mechanical waves) Phys272 - Spring 14 - von Doetinchem - 285 Electricity, magnetism, and light ● ● ● ● Maxwell's equation relate magnetic and electric field Moving charges produce both electric and magnetic fields An electromagnetic wave is formed when a charge accelerates We never spoke about how fast a magnetic field can be measured at a certain distance after a charge starts moving: – Electromagnetic waves do not travel with infinite speed Phys272 - Spring 14 - von Doetinchem - 286 Generating electromagnetic radiation ● ● Look at oscillating L-C circuit Transformation of the LC circuit into a conducting rod: → this is still an oscillating LC circuit Phys272 - Spring 14 - von Doetinchem - 287 Generating electromagnetic radiation ● What do the electric and magnetic fields look like: ● In initial configuration fields are contained in capacitor and inductor ● In the rod configuration the electric and magnetic field overlap ● Current is moving up and down the rod → charge in “capacitor” and current in “inductor” are changing with time → electric and magnetic field change with time → electric and magnetic field propagate with finite velocity → electromagnetic wave Phys272 - Spring 14 - von Doetinchem - 288 Hertzian dipole Source: http://de.wikipedia.org/wiki/Hertzscher_Dipol Change of electric field over time Phys272 - Spring 14 - von Doetinchem - 289 Plane electromagnetic waves ● ● Make the following assumption: – Electric and magnetic field configuration with wave-like behavior – Electric field has only a y component – Magnetic field has only a z component – Both move in x direction with velocity c We have to test if this assumption is consistent with Maxwell's equation Phys272 - Spring 14 - von Doetinchem - 290