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Unformatted text preview: Physics 212
Ph
Lecture 10 Today's Concept:
Kirchhoff’s Rules
Circuits with resistors & batteries Physics 212 Lecture 10, Slide 1 Music
Music
Who is the Artist?
A)
B)
C)
D)
E) Norah Jones
Diana Krall
Jane Monheit
Nina Simone
Marcia Ball • A little gender equality
• Anyone know the connection between Oscar Peterson and Diana Krall?
Krall?
• Both great Canadian jazz pianists – Peterson was one of Krall’s
mentors
Physics 212 Lecture 10 Your
Your Comments:
“I usually can't RESIST putting some lecture thoughts down, but
usually can RESIST putting some lecture thoughts down but
CURRENTLY
CURRENTLY I need to DROP in my bed to GAIN some sleep. ;)”
“Resistance Is Futile! All voltages will be assimilated. your individual
Is Futile! All voltages will be assimilated your individual
currents will be added to our own.”
“I thought I had this, now you lost me. I don't
understand
understand how to figure out where the currents
are going.” We’ll start simply, with
the checkpoints, then to
a calculation “h
“how current flows and direction, how many
fl
di
currents can there be”
“The example with the wire connecting A and B
between the resistors was confusing. I worked it
out on paper and it seems like there should be a
flow
flow.” Thi
This is a great example
– we’ll do it at the end
to make sure of the
concepts “Saw an Explosions in the Sky concert after the exam last night. It was
Physics 212 Lecture 10, Slide
electrifying.” 3 Today’s
Today’s Plan:
• Summary of Kirchoff’s rules – these are the key
concepts
concepts
• Example problem
• Review Checkpoints Physics 212 Lecture 10, Slide 4 Last Time
Resistors in series:
Current through is same.
Voltage drop across is IRi Reffective R1 R2 R3 ... Resistors in parallel:
Voltage drop across is same. 1 Current through is V/Ri Reffective Solved Circuits
R1 V 11
1 ...
R1 R2 R3 R2 = R3 V I1234 R1234 R4
Physics 212 Lecture 10, Slide 5
5 New Circuit
R1
R3
V1 V2 R2 How Can We Solve This One?
R1
V1 V2 R3
R2 = V I1234 R12 THE ANSWER: Kirchhoff’s Rules
Physics 212 Lecture 10, Slide 6
5 Kirchoff’s Voltage Rule V i 0 Kirchoff's Voltage Rule states that the sum of the voltage
changes caused by any elements (like wires, batteries, and
resistors) around a circuit must be zero. WHY?
WHY?
The potential difference between a point and itself is zero ! Physics 212 Lecture 10, Slide 7 Kirchoff’s Current Rule I in I out Kirchoff's Current Rule states that the sum of all
currents
currents entering any given point in a circuit must equal
the sum of all currents leaving the same point. WHY?
WHY?
Electric
Electric Charge is Conserved Physics 212 Lecture 10, Slide 8 Checkpoint
Checkpoint 1
How many potentially different currents are there in the circuit shown?
many potentially different currents are there in the circuit shown? A. 3 B. 4 C. 5 D. 6 E. 7 “There is only one node, splitting the initial current
into two separate currents, meaning that there are 3
separate currents in the circuit.”
A
“The original+ 2 split+1 when they join back
B
together=4”
C
“Electric D
potential decreases in every resistor.” Physics 212 Lecture 10, Slide 9 Checkpoint
Checkpoint 1
How many potentially different currents are there in the circuit shown?
many potentially different currents are there in the circuit shown? I1 I3
I2 I1 A. 3 B. 4 C. 5 I3 D. 6 E. 7 Look at the nodes!
Top node: I1 flows in, I2 and I3 flow out
A
B
Bottom
C
D node: I2 and I3 flow in, I1 flows out That
That’s all of them!
Physics 212 Lecture 10, Slide 10 Checkpoint
Checkpoint 2 GAIN In the following circuit, consider the loop abc. The direction of the current
through each resistor is indicated by black arrows.
DROP If we are to write Kirchoff's voltage equation for this loop in the clockwise direction
starting from point a, what is the correct order of voltage gains/drops that we will
encounter for resistors R1, R2 and R3?
A
60 drop, gain, gain
A. drop, drop, drop
B. gain, gain, gain
C.
B gain, drop, drop
D.
E. drop, drop, gain
50
C
D
40
E
30 20
10 Physics 212 Lecture 10, Slide 11
0 Checkpoint
Checkpoint 2 GAIN In the following circuit, consider the loop abc. The direction of the current
through each resistor is indicated by black arrows.
DROP If we are to write Kirchoff's voltage equation for this loop in the clockwise direction
starting from point a, what is the correct order of voltage gains/drops that we will
encounter for resistors R1, R2 and R3?
A
A. drop, drop, drop
B. gain, gain, gain
C. drop, gain, gain
B gain, drop, drop
D.
E. drop, drop, gain
C
D
E With the current
Against the current VOLTAGE DROP
VOLTAGE GAIN Physics 212 Lecture 10, Slide 12 1
2
1 Calculation
Calculation 2V
1V I2 In this circuit, assume Vi and Ri are known.
What is I2 ?? 1V • Conceptual Analysis:
– Circuit behavior described by Kirchhoff’s Rules:
• KVR: Vdrops = 0
• KCR: Iin = Iout •Strategic Analysis
–
–
– Write down Loop Equations (KVR)
Write down Node Equations (KCR)
Solve Physics 212 Lecture 10, Slide 13 V1 R1
++  + + R3
  V2 R2 I2 In this circuit, assume Vi and Ri are known.
What is I2 ??  V3
++ I1 Calculation
Calculation I3  (1) Label and pick directions for each current
(2) Label the + and – side of each element
This is easy for batteries
For resistors, the “upstream” side is +
Now write down loop and node equations Physics 212 Lecture 10, Slide 14 V1 R1
++  + + R3
  V2 R2 I2  V3
++ I1 Calculation
Calculation
In this circuit, assume Vi and Ri are known.
What is I2 ?? I3  • How many equations do we need to write down in order to solve for I2?
(A) 1
• Why??
–
– (B) 2 (C) 3 (D) 4 (E) 5 We have 3 unknowns: I1, I2, and I3
We need 3 independent equations to solve for these unknowns (3) Choose loops and directions
Physics 212 Lecture 10, Slide 15 V1 R1
++  + + R3
  V2 R2 I2  V3
++ I1 Calculation
Calculation I3  In this circuit, assume Vi and Ri are known.
What is I2 ?? • Which of the following equations is NOT correct?
(A)
(B)
(C)
(D) I 2 = I1 + I 3
 V1 + I1R1  I3R3 + V3 = 0
 V3 + I3R3 + I2R2 + V2 = 0
 V2 – I2R2 + I1R1 + V1 = 0 • Why??
–
– (4) Write down voltage drops
(5) Write down node equation (D) is an attempt to write down KVR for the top loop
Start at negative terminal of V2 and go clockwise
• Vgain (V2) then Vgain (I2R2) then Vgain (I1R1) then Vdrop (+V1)
Physics 212 Lecture 10, Slide 16 R1 V1 R2 V2 R3 V3 I1 Calculation
Calculation I2 In this circuit, assume Vi and Ri are known. I3 • We have the following 4 equations:
1.
2.
3.
4. I2 = I1 + I 3
 V1 + I1R1  I3R3 + V3 = 0
 V3 + I3R3 + I2R2 + V2 = 0
 V2 – I2R2  I1R1 + V1 = 0 • Why??
–
– – What is I2 ??
We need 3 equations:
Which 3 should we use?
A)
A) Any 3 will do
B) 1, 2, and 4
C) 2, 3, and 4 We need 3 INDEPENDENT equations
Equations 2, 3, and 4 are NOT INDEPENDENT
• Eqn 2 + Eqn 3 =  Eqn 4
We must choose Equation 1 and any two of the remaining ( 2, 3, and 4)
Physics 212 Lecture 10, Slide 17 R1 V1 R2 V2 R3 R
2R
R V3 2V
V
V I1
I2
I3 I1
I2
I3 Calculation
Calculation
In this circuit, assume Vi and Ri are known.
assume
What is I2 ??
• We have 3 equations and 3 unknowns.
I2 = I1 + I 3
V1 + I1R1  I3R3 + V3 = 0
V2 – I2R2  I1R1 + V1 = 0 (6) Solve the equations
•The solution will get very messy!
Simplify: assume V2 = V3 = V
V1 = 2V
R1 = R3 = R
R2 = 2R Physics 212 Lecture 10, Slide 18 Calculation:
Calculation: Simplify
In this circuit, assume V and R are known.
R
2R
R 2V
V
V I1
I2 What is I2 ?? • We have 3 equations and 3 unknowns.
I2 = I1 + I 3
2V + I1R  I3R + V = 0 (outside)
V – I2(2R)  I1R + 2V= 0 (top) I3
current
direction • With this simplification, you can verify:
I2 = ( 1/5) V/R
I1 = ( 3/5) V/R
I3 = (2/5) V/R Physics 212 Lecture 10, Slide 19 FollowFollowUp
2V R V 2R
a I1 • We know: I2 = ( 1/5) V/R
I1 = ( 3/5) V/R
I3 = (2/5) V/R I2 b R V I3 • Suppose we short R3: (A)
(B)
(C)
(D) Vab
Vab
Vab
Vab remains the same
changes sign
increases
goes to zero What happens to Vab (voltage across R2?)
R
Why?
Redraw: Bottom Loop Equation:
Vab + V – V = 0
Vab = 0 2R
a 2V
V I1
I2 b V
c d I3 Physics 212 Lecture 10, Slide 20 A V B R R Is there a current flowing between A and B ?
A) Yes
B) No
A & B have the same potential
Current flows from battery and splits at A No current flows between A & B
Some current flows down
Some current flows right
Physics 212 Lecture 10, Slide 21 Checkpoint 3a
Consider the circuit shown below. Note that this question is not identical to the similar looking one
I1
I2
you answered in the prelecture. Which of the following best describes the current flowing in the blue wire connecting points a and b?
A. Positive current flows from a to b
B. Positive current flows from b to a
C. No current flows between a and b “current flowing through R is greater than current flowing through 2R, so current
must flow from a to b”
“potential at b is higher than that at a”
“It has equal to the top and bottom of it”
Physics 212 Lecture 10, Slide 22 Checkpoint 3a
Consider the circuit shown below. Note that this question is not identical to the similar looking one
I1
I2
you answered in the prelecture. I1 I I2 I4 I3 Which of the following best describes the current flowing in the blue wire connecting points a and b?
Which of the following best describes theB.urrent flowing in the blue wire connecting points a and b?
of the following best describes
c Positive current flows from b to a
A. Positive current flows from a to b the current flowing in the blue wire connecting points
C. No current flows between a and b I1R – I2 (2R) = 0 I2 = ½ I1 I4R – I3 (2R) = 0 I4 = 2 I 3 a: I1 = I + I3
b: I + I2 = I4 I1  I3 + ½ I1 = 2I3 I1 = 2I3 I = +I3
Physics 212 Lecture 10, Slide 23 Prelecture
Prelecture What is the same? Checkpoint Current flowing in and out of the battery
2R
3
2R
3 What is different? Current flowing from a to b
Physics 212 Lecture 10, Slide 24 I
2/ I
3 R 1/ I
3 a V
2/ I
3 2/ I
3
2
1/ I
3 2R
b R 1/ I
3 V/2
2R 1/ I
3
1/ I
0
3 1/ I
3 2/ I
3 Physics 212 Lecture 10, Slide 25 Consider the circuit shown below. Checkpoint 3b In which case is the current flowing in the blue wire connecting points a and b the largest?
which case is the current flowing in the blue wire connecting points
the largest?
A. Case A
B. Case B
C. They are both the same “since there is less resistance the voltage drop is less. so the current is greater”
“The current will want to avoid the 4R more than it will want to avoid the 2R.”
th 4R
th it
th 2R
“Since the resistance on both top and bottom is the same, the current in a line
'ab'' (in both cases) is zero.”
ab
Physics
Physics 212 Lecture 10, Slide 26 Consider the circuit shown below. Checkpoint 3b
IA c IB c In which case is the current flowing in the blue wire connecting points a and b the largest?
which case is the current flowing in the blue wire connecting points
the largest?
A. Case A
B. Case B
C. They are both the same Current will flow from left to right in both cases
In
In both cases, Vac = V/2
I2R = 2I4R
IA = IR – I2R
= IR – 2I4R IB = IR – I4R
Physics 212 Lecture 10, Slide 27 Model
Model for Real Battery: Internal Resistance +
r
V0 r
R VL V0
R VL Usually can’t supply too much current to the
th
load without voltage “sagging”
Physics 212 Lecture 10, Slide 28 Kirchhoff’s Laws (1) Label all currents
Choose any direction (2) Label +/ for all elements
Current goes +  (for resistors)
Battery signs fixed! (3) Choose loops and directions
Must start on wire, not element. (4) Write down voltage drops
First sign you hit is sign to use. (5) Write down node equations R1
A + I1
 +
V
1 B V2 +
R2
I2
+ +
V3
I3
R3 R5
 + I4
R4
+ +I
5 Iin = Iout (6) Solve set of equations Physics 212 Lecture 10, Slide 29
17 ...
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 Fall '08
 Kim
 Physics

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