Physics 1202: Lecture 15 Today’s Agenda Announcements: –Lectures posted on: –HW assignments, solutions etc. Homework #4:Homework #4: –Due this Friday Midterm 1: –Answers this Friday.

Faraday's Law v B N S v B S N n B B 

Induction Self-Inductance, RL Circuits L/R V t 0 L  X X X X X X X X X

t I 0  R L/R 2L/R VLVL 0 t   on  off 0 -- VLVL t L/R 2L/R t I 0  R

LC Circuits Consider the LC and RC series circuits shown: L C 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!

RC/LC Circuits RC: current decays exponentially C R i Q -i t L C LC: current oscillates i 0 t i Q

LC Oscillations (qualitative)  L C  L C  L C L C 

Energy transfer in a resistanceless, nonradiating LC circuit. The capacitor has a charge Q max 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  of the oscillations (when R=0)? –(it gets more complicated when R finite…and R is always finite) L C

LC Oscillations (quantitative: requires calculus) Begin with the loop rule: Guess solution: (just harmonic oscillator!) where:   determined from equation , Q 0 determined from initial conditions If C fully charged with, Q 0 at t=0,  Procedure: differentiate above form for Q and substitute into loop equation to find  . L C i Q remember:  L = -L  I /  t  C = -Q/C

Review: LC Oscillations Guess solution: (just harmonic oscillator!) where:   determined from equation , Q 0 determined from initial conditions L C i Q 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 (I o ): At any time: The maximum energy stored in the capacitor and in the inductor are the same:

Lecture 15, ACT 1 At t=0 the capacitor has charge Q 0 ; the resulting oscillations have frequency  0. The maximum current in the circuit during these oscillations has value I . –What is the relation between  0 and  2, the frequency of oscillations when the initial charge = 2Q 0 ? (a)  2 = 1/2  0 (b)  2 =  0 (c)  2 = 2  0 1A

Lecture 15, ACT 1 At t=0 the capacitor has charge Q 0 ; the resulting oscillations have frequency  0. The maximum current in the circuit during these oscillations has value I . (a) I  = I  (b) I  = 2 I  (c) I  = 4 I  What is the relation between I  and I , the maximum current in the circuit when the initial charge = 2Q 0 ? 1B

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

Lecture 15, ACT 2a Consider a simple AC circuit with a purely resistive load. For this circuit the voltage source is  = 10V sin (2  50(Hz)t) and R = 5 . What is the average current in the circuit?   R A) 10 AB) 5 AC) 2 AD) √2 AE) 0 A

Chapter 15, ACT 2b Consider a simple AC circuit with a purely resistive load. For this circuit the voltage source is  = 10V sin (2  50(Hz)t) and R = 5 . What is the average power in the circuit?   R A) 0 W B) 20 WC) 10 WD) 10 √2 W

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 15, ACT 3 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