HR31 - Chapter 31 Electromagnetic Oscillations and...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 31 Electromagnetic Oscillations and Alternating Current In this chapter we will cover the following topics: -Electromagnetic oscillations in an LC circuit -Alternating current (AC) circuits with resistors, capacitors, and inductors in series ( RCL circuits) -Resonance in RCL circuits -Power in AC-circuits -Transformers, AC power transmission (31 - 1)
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
L C The circuit shown in the figure consists of a capacitor and an inductor . We give the capacitor an initial charge and then observe how the circuit behaves. The capacitor will d LC C L Q Oscillations ischarge through the inductor resulting in a time dependent current . i We will show that the charge on the capacitor plates as well as the current 1 in the inductor oscillate with constant amplitude at an angular frequency The total energy in the circuit is t qi LC U ω= 22 he sum of the energy stored in the electric field of the capacitor and the energy stored the magnetic field of the inductor. . The total energy of the circuit does not chan E B EB U U qL i UU U C =+= + ge with time. Thus 0 1 0. 0 dU dt dU q dq di dq di d q d q Li i L q dt C dt dt dt dt dt dt C = =+ = = = + = (31 - 2) 2 2 1 0 dq Lq dt C +=
Background image of page 2
L C 2 2 2 2 1 0 ( ) This is a homogeneous, second order, linear differential equation which we have encountered previously. We 1 0 use d it to describe the simple harmonic dq q dt L Lq d C tC ⎛⎞ += ⎜⎟ +→ = eqs.1 2 2 2 oscillator (SHO) () with solution: ( ) cos( ) dx x dt xt X t ω ωφ =+ eqs.2 If we compare eqs.1 with eqs.2 we find that the solution to the differential equation that describes the -circuit (eqs.1) is: 1 ( ) cos where , and is the phase angle. The current LC qt Q t LC φ = sin Q and depend on the inital conditions: (0) and (0) dq iQ t dt Qi ωω == + Note : co s Q t 1 LC ω= (31 - 3)
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
L C () 22 2 2 2 2 The energy stored in the electric field of the capacitor cos The energy stored in the magnetic field of the inductor sin sin 2 The total energy 2 E B EB qQ Ut CC Li L Q Q t C UU U Q U ωφ ω == + + = + =+ = ()() 2 cos sin 2 The total energy is constant; Q tt ⎛⎞ ⎡⎤ ++ + = ⎜⎟ ⎣⎦ ⎝⎠ energy is conserved 2 2 3 The energy of the has a value of at 0, , , ,... 2 35 The energy of the has a value of at , , 24 4 4 When is maximum is ze QT T tT C T T t C = = electric field maximum magnetic field maximum Note : ro, and vice versa (31 - 4)
Background image of page 4
0 t = 1 2 /8 tT = 3 /4 = 4 3/ 8 = 5 5 /2 = 4 3 2 1 6 6 5/ 8 = 4 = 7/ 8 = 7 8 7 8 (31 - 5)
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 2 If we add a resistor in an RL cicuit (see figure) we must modify the energy equation because now energy is being dissipated on the resistor. 2 EB dU iR dt q UU U C =− =+= + Damped oscillations in an RCL circuit 2 2 2 Li dU q dq di Li i R dt C dt dt →= + = () 2 2 2 /2 2 22 1 0 This is the same equation as that of the damped harmonics o which h scillator: The a as the solution co ngul r f s a : bt m m dq d i dq iL R q dt dt dt dt d dx d x mb k x dt dt xt xe t tC ωφ ++ = =+ =→ = + + = 2 2 2 2 1 requency For the damped circuit the solution is: The angular frequency cos 4 4 Rt L R q RC tQ e t kb LC L mm L ω ′′ = (31 - 6)
Background image of page 6
/2 Rt L Qe Rt L Qe () qt Q Q ( ) cos Rt L Qe t ω φ = + 2 2 1 4 R LC L =− 2 2
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/18/2011 for the course PHY 108 taught by Professor Iashvili during the Fall '08 term at SUNY Buffalo.

Page1 / 28

HR31 - Chapter 31 Electromagnetic Oscillations and...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online