Physics 1202: Lecture 12 Today’s Agenda Announcements: Team problems start this Thursday Team 1: Hend Ouda, Mike Glinski, Stephanie Auger Team 2: Analiese Bruder, Kristen Dean, Alison Smith Office hours: Monday 2:30-3:30 Thursday 3:00-4:00 Homework #5: due this coming Friday Midterm 1: Thursday March 1st (in class) Review session Tuesday Feb. 27 (+ Team problems) Midterm sample + To-Know sheet on web already Chapter 24: AC circuits AC voltage, current + phaser and RMS values C & L in AC circuits + RC & RL circuits RLC circuits resonances 1

24-AC Current L C ~ e R w

Phasors R: V in phase with i Þ C: V lags i by 90° Þ L: V leads i by 90° Þ A phasor is a vector whose magnitude is the maximum value of a quantity (eg V or I) and which rotates counterclockwise in a 2-d plane with angular velocity w. Recall uniform circular motion: y y The projections of r (on the vertical y axis) execute sinusoidal oscillation. w x

Suppose: Phasors for L,C,R i i wt w ß i i wt w i i wt w

w - dependence in AC Circuits The maximum current & voltage are related via the impedence Z Currents AC-circuits as a function of frequency:

24-5 RLC Circuits Phasor diagram Follow the loop Total V useful to analyze an RLC circuit Follow the loop VR = R Imax (in phase) VL= XL Imax (leads by 90o) VC= XC Imax (lags by 90o) Total V V= VR + VL+ VC = Z Imax

Phasors: LCR Ohms -> The phasor diagram has been relabeled in terms of the reactances defined from: Ohms -> The unknowns (Imax,f) can now be solved for graphically since the vector sum of the voltages VL + VC + VR must sum to the driving emf e.

Phasors:LCR f Imax R Imax (XL-XC) Vmax = Imax Z Þ

24-5 RLC Circuits The phase angle for an RLC circuit is: If XL = XC, the phase angle is zero, and the voltage and current are in phase. The power factor:

Phasors:Tips This phasor diagram was drawn as a snapshot of time t=0 with the voltages being given as the projections along the y-axis. Sometimes, in working problems, it is easier to draw the diagram at a time when the current is along the x-axis (when i=0). f ImaxR ImaxXL ImaxXC Vmax “Full Phasor Diagram” From this diagram, we can also create a triangle which allows us to calculate the impedance Z: “ Impedance Triangle” Z |  R | XL-XC |

24-5 RLC Circuits At high frequencies, the capacitive reactance is very small, while the inductive reactance is very large. The opposite is true at low frequencies.

Lecture 12, ACT 1 (c) (a) (b) t=0 i f t=0 A series LCR circuit driven by emf e = e0sinwt produces a current i=imsin(wt-f). The phasor diagram for the current at t=0 is shown to the right. At which of the following times is VC, the magnitude of the voltage across the capacitor, a maximum? (a) (b) (c) i t=0 t=tb t=tc

24-6: Resonance - LC Circuits Consider the LC and RC series circuits shown: L C R Suppose that the circuits are formed at t=0 with the capacitor C charged to a value Q. Claim is that there is a qualitative difference in the time development of the currents produced in these two cases. Why?? Consider from point of view of energy! In the RC circuit, any current developed will cause energy to be dissipated in the resistor. In the LC circuit, there is NO mechanism for energy dissipation; energy can be stored both in the capacitor and the inductor!

current decays exponentially RC/LC Circuits L C LC: current oscillates i t Q +++ - - - RC: current decays exponentially C R i Q -i t 1 +++ - - -

LC Oscillations (qualitative) ß L C Þ L C + - L C Ý Ü L C + -

Energy transfer in a resistanceless, nonradiating LC circuit. The capacitor has a charge Qmax at t = 0, the instant at which the switch is closed. The mechanical analog of this circuit is a block–spring system.

LC Oscillations (quantitative) What do we need to do to turn our qualitative knowledge into quantitative knowledge? • What is the frequency w of the oscillations (when R=0)? L C + - The rms voltages across the capacitor and inductor must be the same; therefore, we can calculate the resonant frequency.

LC Oscillations (quantitative: requires calculus) + - i Q Begin with the loop rule: eC= -Q/C eL= -L DI / Dt Guess solution: (just harmonic oscillator!) remember: where: • w0 determined from equation • f, Q0 determined from initial conditions If C fully charged with , Q0 at t=0, f=0. Procedure: differentiate above form for Q and substitute into loop equation to find w0.

Review: LC Oscillations + - i Q Guess solution: (just harmonic oscillator!) where: • w0 determined from equation • f, Q0 determined from initial conditions which we could have determined from the mass on a spring result:

The energy in LC circuit conserved ! When the capacitor is fully charged: When the current is at maximum (Io): The maximum energy stored in the capacitor and in the inductor are the same: At any time:

Lecture 12, ACT 2 (a) w2 = 1/2 w0 (b) w2 = w0 (c) w2 = 2 w0 At t=0 the capacitor has charge Q0; the resulting oscillations have frequency w0. The maximum current in the circuit during these oscillations has value I0 . What is the relation between w0 and w2 , the frequency of oscillations when the initial charge = 2Q0 ? (a) w2 = 1/2 w0 (b) w2 = w0 (c) w2 = 2 w0

Lecture 12, ACT 2 At t=0 the capacitor has charge Q0; the resulting oscillations have frequency w0. The maximum current in the circuit during these oscillations has value I0 . What is the relation between I0 and I2 , the maximum current in the circuit when the initial charge = 2Q0 ? 1B (a) I2 = I0 (b) I2 = 2 I0 (c) I2 = 4 I0

Resonance For fixed R,C,L the current im will be a maximum at the resonant frequency w0 which makes the impedance Z purely resistive. ie: reaches a maximum when: XL=XC the frequency at which this condition is obtained is given from: Þ Note that this resonant frequency is identical to the natural frequency of the LC circuit by itself! At this frequency, the current and the driving voltage are in phase!

24-6 Resonance in Electrical Circuits In an RLC circuit with an ac power source, the impedance is a minimum at the resonant frequency: XL=XC

Resonance The current in an LCR circuit depends on the values of the elements and on the driving frequency through the relation “ Impedance Triangle” Z |  R | XL-XC | 1 2 x im 2wo w R=Ro m / R0 Suppose you plot the current versus w, the source voltage frequency, you would get: R=2Ro

Power in LCR Circuit The power supplied by the emf in a series LCR circuit depends on the frequency w. It will turn out that the maximum power is supplied at the resonant frequency w0. The instantaneous power (for some frequency, w) delivered at time t is given by: Remember what this stands for The most useful quantity to consider here is not the instantaneous power but rather the average power delivered in a cycle. To evaluate the average on the right, we first expand the sin(wt-f) term.

Power in LCR Circuit Expanding, Taking the averages, +1 sinwtcoswt (Integral of Product of even and odd function = 0) sinwtcoswt wt 2p +1 -1 Generally: Putting it all back together again, sin2wt wt 2p +1 -1 1/2

Power in LCR Circuit The power can be expressed in term of i max: Þ This result is often rewritten in terms of rms values: Þ Power delivered depends on the phase, f, the “power factor” phase depends on the values of L, C, R, and w

Fields from Circuits? We have been focusing on what happens within the circuits we have been studying (eg currents, voltages, etc.) What’s happening outside the circuits?? We know that: charges create electric fields and moving charges (currents) create magnetic fields. Can we detect these fields? Demos: We saw a bulb connected to a loop glow when the loop came near a solenoidal magnet. Light spreads out and makes interference patterns. Do we understand this?

Application of Resonance Tuning a radio A varying capacitor changes the resonance frequency of the tuning circuit in your radio to match the station to be received Metal Detector The portal is an inductor, and the resonant frequency is set to a condition with no metal present When metal is present, it changes the effective inductance, which changes the current The change in current is detected and an alarm sounds

Problem Solution Method: Five Steps: Focus on the Problem - draw a picture – what are we asking for? Describe the physics what physics ideas are applicable what are the relevant variables known and unknown Plan the solution what are the relevant physics equations Execute the plan solve in terms of variables solve in terms of numbers Evaluate the answer are the dimensions and units correct? do the numbers make sense?

Recap of Today’s Topic : Announcements: Team problems start this Thursday Team 1: Hend Ouda, Mike Glinski, Stephanie Auger Team 2: Analiese Bruder, Kristen Dean, Alison Smith Office hours: Monday 2:30-3:30 Thursday 3:00-4:00 Homework #5: due this coming Friday Midterm 1: Thursday March 1st (in class) Review session Tuesday Feb. 27 (+ Team problems) Midterm sample + To-Know sheet on web already Chapter 24: AC circuits AC voltage, current + phaser and RMS values C & L in AC circuits + RC & RL circuits RLC circuits resonances