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Physics 272 April 14 Spring 2015 www.phys.hawaii.edu/~philipvd/pvd_15_spring_272_uhm go.hawaii.edu/KO Prof. Philip von Doetinchem [email protected] PHYS272 - Spring 15 - von Doetinchem - 85 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 15 - von Doetinchem - 86 Resonance in alternating-current circuits minimum of impedance ● A minimum in impedance exists where X L and XC cancel PHYS272 - Spring 15 - von Doetinchem - 87 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 15 - von Doetinchem - 88 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 15 - von Doetinchem - 89 Tailoring an AC circuit https://www.youtube.com/watch?v=ZYgFuUl9_Vs PHYS272 - Spring 15 - von Doetinchem - 90 ● ● ● ● 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 15 - von Doetinchem - 91 Tuning a radio ● Resonance frequency: ● Reactances: PHYS272 - Spring 15 - von Doetinchem - 92 Tuning a radio ● RMS current: ● Voltages: VL, VC are both significantly larger than V R → they cancel at resonance (180deg phase angle) PHYS272 - Spring 15 - von Doetinchem - 93 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 15 - von Doetinchem - 95 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 15 - von Doetinchem - 96 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 15 - von Doetinchem - 97 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 (also using a diode) PHYS272 - Spring 15 - von Doetinchem - 98 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 15 - von Doetinchem - 99 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 15 - von Doetinchem - 100 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 15 - von Doetinchem - 103 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 15 - von Doetinchem - 104 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 15 - von Doetinchem - 105 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 15 - von Doetinchem - 106 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 15 - von Doetinchem - 107 Hertzian dipole http://de.wikipedia.org/wiki/Hertzscher_Dipol#/media/File:DipoleRadiation.gif Change of electric field over time PHYS272 - Spring 15 - von Doetinchem - 108 Plane electromagnetic waves ● ● ● Electromagnetic waves are transverse waves with an electric and a magnetic component → similar to waves on a string 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 different instances in time PHYS272 - Spring 15 - von Doetinchem - 109 Plane electromagnetic waves ● Why perpendicular? ● Faraday's law: ● Integration path perpendicular to electric field: no contribution (1) ● Magnetic flux change: Propagation velocity (2) (1) and (2) area increase in time dt ● Magnetic field component in z direction is crucial to comply to Faraday's law PHYS272 - Spring 15 - von Doetinchem - 110 Plane electromagnetic waves ● ● In a very similar way: use Ampere's law (no conduction current): any components of E and B not perpendicular to each other and not perpendicular to the direction do not play a role to fulfill Maxwell's equation → Electric and magnetic fields must be perpendicular ● To fulfill Maxwell's equations → wave must also fulfill: Propagation velocity PHYS272 - Spring 15 - von Doetinchem - 111 Key properties of electromagnetic waves ● ● Electromagnetic waves fulfill the general wave equation for both electric and magnetic fields (not discussed here): Wave is transverse: – ● Electric field and magnetic field are perpendicular to the direction of the wave In vacuum (or if medium is not changing) the electromagnetic wave is traveling at a constant velocity ● Electromagnetic waves do not need a medium ● Simple plane waves can be generalized – E and B fields can be superposed – For each superposition the simple principles of the plane electromagnetic waves apply – Also wave packets or sinusoidal waves fulfill Maxwell's equations PHYS272 - Spring 15 - von Doetinchem - 112 Sinusoidal electromagnetic waves ● ● ● Sinusoidal waves are essential, e.g., to understand light, X-rays, or radio communication The entire pattern travels in the direction of propagation Electric and magnetic field are still transverse to the propagation direction at any instant snapshot at particular instant of time PHYS272 - Spring 15 - von Doetinchem - 113 Fields of a sinusoidal waves ● ● Electric and magnetic fields oscillate in phase: the fields are at the same time at maximum, zero, and minimum Electromagnetic wave can be described by wave function depending on location and time k is the wave number: 2/ ● Vector product of electric field and magnetic field always points in the propagation direction PHYS272 - Spring 15 - von Doetinchem - 114 Fields of a sinusoidal waves ● ● The electric field and magnetic field amplitudes are related by: The sign in front of t denotes the direction of the wave: – Negative sign: positive x direction – Positive sign: negative x direction PHYS272 - Spring 15 - von Doetinchem - 115 Why are magnetic and electric field in phase? ● ● Assume arbitrary phase angle: Relation between electric and magnetic field using Faraday's law: snapshot at particular instant of time PHYS272 - Spring 15 - von Doetinchem - 116 Why are magnetic and electric field in phase? ● Applying Faraday's law: assumption: variation of BZ between location x and x+x is small at any particular instance in time calculate magnetic flux through area PHYS272 - Spring 15 - von Doetinchem - 117 Why are magnetic and electric field in phase? ● Use our approach for the wave function assuming a phase angle: = ● To make these equations equal at all times we have to require PHYS272 - Spring 15 - von Doetinchem - 118 Electromagnetic waves in matter ● The wave speed changes in matter (we assume a non-conducting dielectric) ● We have to use the dielectric constant and the permittivity of the material: ● ● ● Electromagnetic waves can never travel faster than the speed of light in vacuum Definition of index of refraction n of a material (→ optics) Careful: – dielectric “constant” depends on frequency – for high frequencies materials cannot be polarized as fast as for constant electric fields → reduces dielectric “constant” PHYS272 - Spring 15 - von Doetinchem - 119