Download Electric and Magnetic Oscillations: Inductors and Capacitors and more Study notes Physics in PDF only on Docsity! olnetieteit Buth
Lecture 29
Electrical Oscillations, LC Circuits
03/25/2009
• Notion of electric potential does not work for electric fields produced by induction • Inductance of a solenoid • SI unit henry • Self induction: an EMF appears in any coil in which the current is changing: • Direction of self-induced EMF from Lenz’s law • “Charging” an inductor: • “Discharging” an inductor: € ΦB = NAB = Li dt diLEMF −= € i = E R 1− e − Rt L L Rt e R i − = E • Energy stored in solenoid • Energy density • Although derived for a special case, expression holds generally € UB = 1 2 Li2 = µ0n 2Ali2 2 € L = µ0n 2Al € uB = UB V € uB = µ0n 2i2 2 = B2 2µ0 € B = µ0in Oscillators are very useful in practical applications, for instance, to keep time, or to focus energy in a system. All oscillators operate along the same principle: they are systems that can store energy in more than one way and exchange it back and forth between the different storage possibilities. For instance, in pendulums (and swings) one exchanges energy between kinetic and potential form. potkintot EEE += 22 2 1 2 1 xkvmEtot += + == dt dxxk dt dvvm dt dEtot 2 2 12 2 10 dt dxv = 0=+ xk dt dvm )cos()( :Solution 00 φω += txtx phase : frequency : amplitude : 0 0 φ ω x m k =ω 02 2 =+ xk dt xdm Newton’s law F=ma! -1.5 -1 -0.5 0 0.5 1 1.5 Time Charge Current )cos( 00 φω += tqq )sin( 00 φωω +−== tqdt dqi )(sin 2 1 2 1 0 22 0 22 φωω +== tqLLiEmag 0 0.2 0.4 0.6 0.8 1 1.2 Time Energy in capacitor Energy in coil )(cos 2 1 2 1 0 22 0 2 φω +== tq CC qEele LC xx 1 and ,1sincos that,grememberin And 22 ==+ ω 2 02 1 q C EEE elemagtot =+= The energy is constant and equal to what we started with. • In the circuit shown, the switch is in position “a” for a long time. It is then thrown to position “b.” • Calculate the amplitude of the resulting oscillating current. • Switch in position “a”: charge on capacitor: (1 µF)(10 V) = 10 µC • Switch in position “b”: maximum charge on capacitor: q0 = 10 µC • So, amplitude of oscillating current = == )10( )1)(1( 1 0 CFmH q µ µ ω 0.316 A b a E=10 V 1 mH 1 µF )cos( 00 φω += tqq )sin( 00 φωω +−== tqdt dqi • Energy stored in a magnetic field • Energy density in magnetic field
