Physics 1202: Lecture 16 Today’s Agenda Announcements: –Lectures posted on: –HW assignments, etc. Homework #5:Homework #5: –Due next Friday Midterm 1: –Answers today –New average = 63%

L C   R 

 R Circuit We begin by considering simple circuits with one element (R,C, or L) in addition to the driving emf. Begin with R: Loop eqn gives:  Voltage across R in phase with current through R   i R R Note: this is always, always, true… always. 0 t x mm  m 0 0 t  m / R  m / R 0

RMS Values Average values for I,V are not that helpful (they are zero). Thus we introduce the idea of the Root of the Mean Squared. In general, So Average Power is,

 C Circuit (… calculus !) Now consider C: Loop eqn gives:    C  Voltage across C lags current through C by one-quarter cycle (90  ). Is this always true? YES 0 t x mm  m 0 t 0 0  C  m  C  m

Lecture 16, ACT 1 A circuit consisting of capacitor C and voltage source  is constructed as shown. The graph shows the voltage presented to the capacitor as a function of time. –Which of the following graphs best represents the time dependence of the current i in the circuit? (a) (b) (c) i t i t t i  t

 L Circuit (… calculus !) Now consider L: Loop eqn gives:   Voltage across L leads current through L by one- quarter cycle (90  ).   L Yes, yes, but how to remember? 0 t x mm  m 0 t x  m  L  m  L 0 0

Phasors 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 . Recall uniform circular motion: The projections of r (on the vertical y axis) execute sinusoidal oscillation.    R: V in phase with i C: V lags i by 90  L: V leads i by 90   x y y

0 i 0 i Phasors for L,C,R i tt  i tt  i tt   Suppose: 0 i

A series LCR circuit driven by emf  =  0 sin  t produces a current i=i m sin(  t-  ). The phasor diagram for the current at t=0 is shown to the right. –At which of the following times is V C, the magnitude of the voltage across the capacitor, a maximum? Lecture 16, ACT 2 i  t=0 (a) (b) (c) i t=0 i t=t b i t=t c

Series LCR AC Circuit Consider the circuit shown here: the loop equation gives: Here all unknowns, (i m,  ), must be found from the loop eqn; the initial conditions have been taken care of by taking the emf to be:  m sin  t. To solve this problem graphically, first write down expressions for the voltages across R,C, and L and then plot the appropriate phasor diagram. L C   R Assume a solution of the form:  C = -Q/C  L = -L  I /  t  R = -RI

Phasors: LCR Assume: From these equations, we can draw the phasor diagram to the right. L C   R  This picture corresponds to a snapshot at t=0. The projections of these phasors along the vertical axis are the actual values of the voltages at the given time. Given:   imim 

Phasors: LCR The phasor diagram has been relabeled in terms of the reactances defined from:  L C   R The unknowns (i m,  ) can now be solved for graphically since the vector sum of the voltages V L + V C + V R must sum to the driving emf .

Phasors:LCR  

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. y x  imRimR imXLimXL imXCimXC mm “Full Phasor Diagram” From this diagram, we can also create a triangle which allows us to calculate the impedance Z: 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). “ Impedance Triangle” Z |  R | X L -X C |

Resonance For fixed R,C,L the current i m will be a maximum at the resonant frequency  0 which makes the impedance Z purely resistive. the frequency at which this condition is obtained is given from:  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! ie: reaches a maximum when: X L =X C

Resonance The current in an LCR circuit depends on the values of the elements and on the driving frequency through the relation Suppose you plot the current versus , the source voltage frequency, you would get: “ Impedance Triangle” Z |  R | X L -X C | 12 x imim 0 0  o  R=R o  m / R 0 R=2R o

Power in LCR Circuit The power supplied by the emf in a series LCR circuit depends on the frequency . It will turn out that the maximum power is supplied at the resonant frequency  0. The instantaneous power (for some frequency,  ) delivered at time t is given by: 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(  t-  ) term. Remember what this stands for

Power in LCR Circuit Expanding, Taking the averages, Generally: sin 2  t tt 0  0 +1 Putting it all back together again, 0 1/2 (Integral of Product of even and odd function = 0) sin  tcos  t tt 0  0 +1

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

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?