Physics 1202: Lecture 11 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 by end of week Chapter 23: induction Inductance of solenoid + in series & parallel Chapter 24: AC circuits AC voltage, current + phaser and RMS values C & L in AC circuits + RC & RL circuits 1

23-Induction Faraday's Law n B q v B N S

23-7: Inductance Consider the loop at the right. X X X X switch closed Þ current starts to flow in the loop. \ magnetic field produced in the area enclosed by the loop. \ flux through loop changes \ emf induced in loop opposing initial emf Self-Induction: the act of a changing current through a loop inducing an opposing current in that same loop.

Inductance I The magnetic field produced by the current in the loop shown is proportional to that current. The flux, therefore, is also proportional to the current. We define this constant of proportionality between flux and current to be the inductance L. We can also define the inductance L, using Faraday's Law, in terms of the emf induced by a changing current.

23-7: Inductance I So, the inductance is The magnetic field produced by the current in the loop shown is proportional to that current. The flux, therefore, is also proportional to the current. So, the inductance is

Calculation l For a single turn, N turns (n: number of turns per unit length) Long Solenoid: N turns total, radius r, Length l For a single turn, The total flux through solenoid is given by: Inductance of solenoid can then be calculated as: This (as for R and C) depends only on geometry (material)

23-9: Energy of an Inductor How much energy is stored in an inductor when a current is flowing through it? a b L I Start with loop rule: eR= -RI eL= -L DI / Dt Multiply this equation by I: PL is the rate at which energy is being stored in the inductor: Þ I i DI Li LiDI LI The total U stored in the inductor when the current = I is the shaded triangle: Þ

Where is the Energy Stored? Claim: (without proof) energy is stored in the Magnetic field itself (just as in the Capacitor / Electric field case). To calculate this energy density, consider the uniform field generated by a long solenoid: l r N turns The inductance L is: Energy U: We can turn this into an energy density by dividing by the volume containing the field:

23-10: Transformers eP eS Þ \ Device to change e (or the voltage) 2 coils wrapped around iron core Primary (P) and secondary (S) B-field inside core Time varying current in P (with NP) Faraday's Law: eP = - D Dt B F NP eS = - D Dt B F NS Time varying flux induces emf in secondary coil S (with NS) Same varying flux Þ \

Inductors in series & parallel V L1 L2 Like resistors: basically wires … Inductors in series the current is the same in both L1 and L2 the voltage drops add Inductors in parallel the voltage drop is the same in both L1 and L2 the currents add V L1 L2

24-AC Current L C ~ e R w

24-AC Current Nikola Tesla 1865 – 1943 Inventor Key figure in development of AC electricity High-voltage transformers Transport of electrical power via AC transmission lines Beat Edison’s idea of DC transmission lines 24-AC Current

24-1: Alternating voltage and current We begin by considering simple circuits with one element (R,C, or L) in addition to the driving emf. ~ e i R Begin with R: Loop eqn gives: Þ Voltage across R in phase with current through R Note: this is always, always, true… always. t x m m t m / R m / R

© 2017 Pearson Education, Inc. 24-1 Phaser To visualize the phase relationships between V & I Phasors vectors whose length is the maximum V or I, rotate around an origin with the angular speed of the oscillating current. The instantaneous value of V or I Projection of phasor on the y axis. Like circular motion in 1201 © 2017 Pearson Education, Inc.

RMS Values Average values for I,V are not that helpful they are zero But power is not ! Recall analogy with fluid t x m m m / R m / R

RMS Values Thus we introduce the idea of the Root of the Mean Squared. In general, So Average Power is. 120 volts is the rms value of household ac

Lecture 11, ACT 1a ~ e R Consider a simple AC circuit with a purely resistive load. For this circuit the voltage source is e = 10V sin (2p50(Hz)t) and R = 5 W. What is the average current in the circuit? A) 10 A B) 5 A C) 2 A D) √2 A E) 0 A

Chapter 11, ACT 1b ~ e R Consider a simple AC circuit with a purely resistive load. For this circuit the voltage source is e = 10V sin (2p50(Hz)t) and R = 5 W. What is the average power in the circuit? A) 0 W B) 20 W C) 10 W D) 10 √2 W

24-2: Capacitors in AC Circuits Now consider C: Loop eqn gives: ~ e C Þ Þ Voltage across C lags current through C by one-quarter cycle (90°). Is this always true? YES t x m m t Cm Cm

24-2: Capacitors in AC Circuits Since the rms current in the capacitor is related to its capacitance and to the frequency

24-2 Capacitive reactance In analogy with resistance, we write: Þ with

24-2 Phaser for C V and I in a capacitor are not in phase: V lags by 90º t x m m Cm Cm

Lecture 11, ACT 2 (a) (b) (c) e A circuit consisting of capacitor C and voltage source e 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

24-3 RC Circuits Current I across R and C This phasor diagram are not in phase maximum I is not the sum of the maximum of IR and the maximum of IC they do not peak at the same time. This phasor diagram V across C and across R are at 90º in the diagram; if added as vectors, we find the maximum

24-3 RC Circuits This has the exact same form as V = IR : if we define the impedance, Z:

24-3 RC Circuits There is a phase angle between the voltage and the current, as seen in the diagram. The power in the circuit is given by: Because of this, the factor cos φ is called the power factor.

24-4: inductors in AC Circuits Now consider L: Loop eqn gives: ~ e L Þ Þ Voltage across L leads current through L by one- quarter cycle (90°). Yes, yes, but how to remember? t x m m t x m L m L

24-4: inductors in AC Circuits Since the rms current in the capacitor is related to its capacitance and to the frequency © 2017 Pearson Education, Inc.

24-4 Inductive reactance In analogy with resistance, we write: Þ with Just as with capacitance, we can define inductive reactance:

24-4 Phaser for L V and I in an inductor are not in phase: V leads by 90º t x m m m L m L

RL Circuits Current I across R and L This phasor diagram are not in phase maximum I is not the sum of the maximum of IR and the maximum of IL they do not peak at the same time. This phasor diagram V across L and across R are at 90º in the diagram; if added as vectors, we find the maximum

24-4 RL Circuits This has the exact same form as V = IR : if we define the impedance, Z:

24-4 RL Circuits As before, there is a phase angle between V and I The power in the circuit is Because of this, the factor cos φ is called the power factor.

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:

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 by end of week Chapter 23: induction Inductance of solenoid + in series & parallel Chapter 24: AC circuits AC voltage, current + phaser and RMS values C & L in AC circuits + RC & RL circuits