1. 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

2. 23-Induction
Faraday's Law n B q v B N S

3. 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.

4. 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.

5. 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

6. 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)

7. 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

8. 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:

9. 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 \

10. Inductors in series & parallelV 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

11. 24-AC Current L C ~ e R w

12. 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

13. 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

14. © 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
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15. 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

16. 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

17. 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

18. 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

19. 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

20. 24-2: Capacitors in AC Circuits
Since
the rms current in the capacitor is related to its capacitance and to the frequency

21. 24-2 Capacitive reactance
In analogy with resistance, we write: Þ
with

22. 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

23. 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. 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

25. 24-3 RC Circuits This has the exact same form as V = IR :
if we define the impedance, Z:

26. 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.

27. 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

28. 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.

29. 24-4 Inductive reactance In analogy with resistance, we write: Þ with
Just as with capacitance, we can define inductive reactance:

30. 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

31. 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

32. 24-4 RL Circuits This has the exact same form as V = IR :
if we define the impedance, Z:

33. 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.

34. 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

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

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

37. 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?

38. 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