1. Physics 1202: Lecture 6 Today’s Agenda
Announcements:
Lectures posted on:
Office hours:
Monday 2:30-3:30
Thursday 3:00-4:00
Homework #2: due this coming Friday/
Labs: Already begun last week
Policy on clicker questions
80 % of total points gives 100%
No make-up for missed clicker questions …
Policy on Homework
Lowest homework will be dropped
No extension 1

2. Today’s Topic : Chapter 21: Electric current & DC-circuits Review
Resistance and Ohm’s law
Power
Resistance in series & parallel
Kirchhoff’s rules
Capacitances in circuits
In series & parallel
RC-circuits
Measuring devices

3. 21- Electric Current e R I
21-1: Electric current
I = DQ / Dt
 = R I

4. 21-2: Resistance & Ohm’s LawV I R
Resistance
Resistance is defined to be the ratio of the applied voltage to the current passing through.
UNIT: OHM = W
What does it mean ?
it is the a measure of the friction slowing the motion of charges
Analogy with fluids

5. 21-3: Energy & Power Batteries & Resistors Energy expended
chemical
to electrical
to heat
Rate is:
What’s happening?
Assert:
Charges per time
Energy “drop” per charge
For Resistors:
Units okay?

6. 21-4: Electric Circuits e R I
 = R I

7. Devices
Conductors:
Purpose is to provide zero potential difference between 2 points.
Electric field is never exactly zero.. All conductors have some resistivity.
In ordinary circuits the conductors are chosen so that their resistance is negligible.
Batteries (Voltage sources, seats of emf):
Purpose is to provide a constant potential difference between 2 points.
Cannot calculate the potential difference from first principles.. electrical « chemical energy conversion. Non-ideal batteries will be dealt with in terms of an "internal resistance". + - V OR

8. Devices R I V Resistors:
Purpose is to limit current drawn in a circuit.
Resistance can be calculated from knowledge of the geometry of the resistor AND the “resistivity” of the material out of which it is made.
The effective resistance of series and parallel combinations of resistors will be calculated using the concepts of potential difference and current conservation (Kirchoff’s Laws). V I R
Resistance
Resistance is defined to be the ratio of the applied voltage to the current passing through.
UNIT: OHM = W

9. Resistors in Series R1 The Voltage “drops”: R2b c R1 R2 I
The Voltage “drops”: a c
Reffective
Whenever devices are in SERIES, the current is the same through both !
This reduces the circuit to:
Hence:

10. Another (intuitive) way...L2 L1
Consider two cylindrical resistors with lengths L1 and L2
Put them together, end to end to make a longer one...

11. Equivalent Resistance – Series: An Example
Four resistors are replaced with their equivalent resistance

12. Resistors in Parallel I R1 R2 I1 I2 V I R V a d a d Þ Þ What to do?
Very generally, devices in parallel have the same voltage drop
But current through R1 is not I ! Call it I1. Similarly, R2 «I2. a d I R V
How is I related to I 1 & I 2 ?? Current is conserved! Þ Þ

13. Another (intuitive) way...
Consider two cylindrical resistors with
cross-sectional areas A1 and A2 V R1 R2 A1 A2
Put them together, side by side … to make a “fatter” one with A=A1+A2 , Þ

14. Summary R1 V R2 V R1 R2 Resistors in series Resistors in parallel
the current is the same in both R1 and R2
the voltage drops add
Resistors in parallel
the voltage drop is the same in both R1 and R2
the currents add V R1 R2

15. Equivalent Resistance – Complex Circuit

16. Lecture 6, ACT 1 V R R
I have two identical light bulbs. First I hook them up in series. Then I hook them up in parallel. In which case are the bulbs brighter? (The resistors represent light bulbs whose brightness is proportional to P = I2R through the resistor.)
A) Series B) Parallel C) The same

17. 21-4:Kirchoff’s Rules e1 R I1 e2 I2 R I3 e3 R

18. Kirchoff's First Rule "Loop Rule" or “Kirchoff’s Voltage Law (KVL)”
"When any closed circuit loop is traversed, the algebraic sum of the changes in potential must equal zero."
KVL:
This is just a restatement of what you already know: that the potential difference is independent of path!
RULES OF THE ROAD:
We will follow the convention that voltage gains enter with a + sign and voltage drops enter with a - sign in this equation. e1 I e1 R1 e2 R2 I R1 R2 e2
Move clockwise around circuit:
= 0 e1
- e2
- IR1
- IR2

19. Loop Example a d b e c f e1 R1 I R2 R3 R4 e2
KVL: Þ Þ

20. Lecture 6, ACT 2 (a) I1 < I0 (b) I1 = I0 (c) I1 > I0
Consider the circuit shown.
The switch is initially open and the current flowing through the bottom resistor is I0.
After the switch is closed, the current flowing through the bottom resistor is I1.
What is the relation between I0 and I1? R
12 V I
(a) I1 < I0
(b) I1 = I0
(c) I1 > I0

21. Kirchoff's Second Rule "Junction Rule" or “Kirchoff’s Current Law (KCL)”
In deriving the formula for the equivalent resistance of 2 resistors in parallel, we applied Kirchoff's Second Rule (the junction rule).
"At any junction point in a circuit where the current can divide (also called a node), the sum of the currents into the node must equal the sum of the currents out of the node."
This is just a statement of the conservation of charge at any given node.

22. Junction Example
Junction: e1 I1 e2 e3 R I2 I3
Outside loop:
Top loop:

23. Chapter 6, ACT 3 (a) (Va -Vd) < (Va -Vc) (b) (Va -Vd) = (Va -Vc)I1 I2 a b d c
50W
20W
80W
Consider the circuit shown:
What is the relation between Va -Vd and Va -Vc ? 3A
(a) (Va -Vd) < (Va -Vc)
(b) (Va -Vd) = (Va -Vc)
(c) (Va -Vd) > (Va -Vc) 3B
What is the relation between I1 and I2?
(a) I1 < I2
(b) I1 = I2
(c) I1 > I2

24. Chapter 6, ACT 4
An ammeter A is connected between points a and b in the circuit below, in which the four resistors are identical. The current through the ammeter is
A) I B) I/ C) I/ D) I/ E) 0

25. 21-6: Circuits Containing Capacitors
Capacitors can also be connected in series or in parallel

26. Capacitors in ParallelV a b Q2 Q1 V a b Q
Find “equivalent” capacitance C in the sense that no measurement at a,b could distinguish the above two situations.
Parallel Combination:
Þ Total charge:
Q = Q1 + Q2
Equivalent Capacitor: Þ
C = C1 + C2

27. º Þ Capacitors in Series a b +Q -Q a b +Q -Q
Find “equivalent” capacitance C in the sense that no measurement at a,b could distinguish the above two situations.
The charge on C1 must be the same as the charge on C2 since applying a potential difference across ab cannot produce a net charge on the inner plates of C1 and C2 .
RHS: Þ
LHS:

28. Examples: Combinations of Capacitors
How do we start??
Recognize C3 is in series with the parallel combination on C1 and C2. i.e. Þ

29. Capacitors in ParallelV a b Q2 Q1 V a b Q Þ
C = C1 + C2
Capacitors in Series a b +Q -Q a b +Q -Q Þ

30. 21-7: RC Circuits Consider the circuit shown:
What will happen when we close the switch ?
Add the voltage drops going around the circuit, starting at point a.
IR + Q/C – V = 0
In this case neither I nor Q are known or constant. But they are related, V a b c R C
This is a simple, linear differential equation.

31. RC Circuits To get Current, I = dQ/dt I Q t t Case 1: Charging
Q1 = 0, Q2 = Q and t1 = 0, t2 = t V a b c R C
To get Current, I = dQ/dt t I Q t

32. RC Circuits To get Current, I = dQ/dt t I Q tV a b c R C
Case 2: Discharging: Q1 = Q0 , Q2 = Q and t1 = 0, t2 = t
To discharge the capacitor we have to take the battery out of the circuit (V=0) c
To get Current, I = dQ/dt Q t t I

33. Chapter 6, ACT 5 V a b c R C
Consider the simple circuit shown here. Initially the switch is open and the capacitor is charged to a potential VO. Immediately after the switch is closed, what is the current ? c
A) I = VO/R B) I = 0 C) I = RC D) I = VO/R exp(-1/RC)

34. 21-8: Electrical Instruments
The Ammeter
The device that measures current is called an ammeter. R1 R2 - A + I e
Ideally, an ammeter should have zero resistance so that
the measured current is not altered.

35. Electrical Instruments
The Voltmeter
The device that measures potential difference
is called a voltmeter. I2 R1 R2 I V Iv e
An ideal voltmeter should have infinite resistance so that
no current passes through it.

36. 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?

37. Example: Power in Resistive Electric Circuits
A circuit consists of a 12 V battery with internal resistance of 2 connected to a resistance of 10 . The current in the resistor is I, and the voltage across it is V. The voltmeter and the ammeter can be considered ideal; that is, their resistances are infinity and zero, respectively.
What is the current I and voltage V measured by those two instruments ? What is the power dissipated by the battery ? By the resistance ? What is the total power dissipated in the circuit ?
Comment on these various powers.

38. Step 1: Focus on the problem
Drawing with relevant parameters
Voltmeter can be put a two places V
What is the question ?
What is I ?
What is V ?
What is Pbattery ?
What is PR ?
What is Ptotal ?
Comment on the various
P’s e R I r A V
10 
2 
12 V

39. Step 2: describe the physics
What concepts are relevant ?
Potential difference in a loop is zero
Energy is dissipated by resistance
What are the known and unknown quantities ?
Known: R = 10 ,r = 2  = 12 V
Unknown: I, V, P’s

40. Step 3: plan the solution
What are the relevant physics equations ?
Kirchoff’s first law:
Power dissipated:
For a resistance

41. Step 4: solve with symbols
Find I:
e - Ir - IR = 0 e R I r A
Find V:
Find the P’s:

42. Step 4: solve numerically
Putting in the numbers

43. Step 5: Evaluate the answers
Are units OK ?
[ I ] = Amperes
[ V ] = Volts
[ P ] = Watts
Do they make sense ?
the values are not too big, not too small …
total power is larger than power dissipated in R
Normal: battery is not ideal: it dissipates energy

44. Recap of today’s lecture
Chapter 21: Electric current & DC-circuits
Electric current
Resistance and Ohm’s law
Power
Resistance in series & parallel
Kirchhoff’s rules
Capacitances in series & parallel
RC-circuits
Measuring devices

45. Today’s Topic : Chapter 21: Electric current & DC-circuits Review
Resistance and Ohm’s law
Power
Resistance in series & parallel
Kirchhoff’s rules
Capacitances in circuits
In series & parallel
RC-circuits
Measuring devices