1. Chapter 32 Fundamentals of Circuits
L.A. Bumm

2. ideal wire with zero resistance
Schematic Symbols
ideal wire with zero resistance

3. Basic Circuits and Kirchhoff’s Loop Law

4. Power in Electrical Circuits
Power supplied by batteries

5. Power in Electrical Circuits
Power dissipated in resistors.
Electrical energy is converted in to heat.
Because current and voltage difference for resistors are related by resistance (Ohm’s Law), we can eliminate I or ΔV from the power equation. But this only works for resistors!!.
This is always true:

6. Clicker Question
Which schematic is NOT equivalent to the cartoon picture? A) B) C) D)

7. These are all statements of Ohm’s Law
Clicker Question
Which equation is not Ohm’s Law
These are all statements of Ohm’s Law A) B) C) D) E)

8. Resistors in Series and Parallel
The current is the same through each resistor.
The voltage drops across each resistor sum to the voltage drop across all the resistors.
The equivalent resistance will be equal to or greater than the largest resistance in the series circuit.
Resistors in Parallel
The voltage drop is the same across each resistor.
The current through each resistor sums to the total current supplied to the parallel circuit.
The equivalent resistance is less than or equal to the smallest resistor in the parallel circuit.

9. Combinations of series and parallel resistors.
Any network of resistors can be reduced to a single equivalent resistance. When the network is not a simple series or parallel combination, it can always be reduced to a single resistor by the sequential application of the series and parallel resistor combination rules. Don’t be afraid to redraw the circuit to make it look more familiar to you.
Reduce this circuit to its equivalent resistance.

10. Example: A nichrome wire with diameter d and length L is connected to a battery of EMF E by two lengths of copper wire with the same d and L as the nichrome wire. Draw a schematic of this circuit. Do not neglect the resistance of the copper wire. What is the resistance of each segment of wire. What is the current in the circuit. How much power is supplied by the battery. How much power is dissipated in the nichrome wire. What percentage of the power is delivered to the nichrome wire? E = 1.5 V; d = 1.0 mm; L = 200 cm; ρnichrome = 1.5×10−6 Ωm; ρCu = 1.7×10−8 Ωm
We are ask to find RNC, RCu, I, Pbat, PNC, and PNC/Pbat. E
RNC I
RCu1
RCu2

11. Example: A nichrome wire with diameter d and length L is connected to a battery of EMF E by two lengths of copper wire with the same d and L as the nichrome wire. Draw a schematic of this circuit. Do not neglect the resistance of the copper wire. What is the resistance of each segment of wire. What is the current in the circuit. How much power is supplied by the battery. How much power is dissipated in the nichrome wire. What percentage of the power is delivered to the nichrome wire? E = 1.5 V; d = 1.0 mm; L = 200 cm; ρnichrome = 1.5×10−6 Ωm; ρCu = 1.7×10−8 Ωm
We are ask to find RNC, RCu, I, Pbat, PNC, and PNC/Pbat. E
RNC I
RCu1
RCu2

12. Example: A nichrome wire with diameter d and length L is connected to a battery of EMF E by two lengths of copper wire with the same d and L as the nichrome wire. Draw a schematic of this circuit. Do not neglect the resistance of the copper wire. What is the resistance of each segment of wire. What is the current in the circuit. How much power is supplied by the battery. How much power is dissipated in the nichrome wire. What percentage of the power is delivered to the nichrome wire? E = 1.5 V; d = 1.0 mm; L = 200 cm; ρnichrome = 1.5×10−6 Ωm; ρCu = 1.7×10−8 Ωm
We are ask to find RNC, RCu, I, Pbat, PNC, and PNC/Pbat. E
RNC I
RCu1
RCu2
We could have ignored the resistance of the copper wire, depending on our application.

13. Kirchhoff’s Laws
Kirchhoff’s Junction Law (Conservation of charge) At any junction point, the sum of all the currents entering the junction must equal the sum of all currents leaving the junction. Kirchhoff’s Loop Law (Conservation of energy) The sum of the changes in potential around any closed path of a circuit must be zero.

14. How to Use the Kirchhoff’s Laws
Draw the circuit and draw currents with an arrow in every separate branch of the circuit. (A branch is a section where the current does not change.)
Apply the junction law to enough junctions so that every current is used at least once.
Apply the loop law to enough closed loops so that each current appears at least once. Remember the sign convention for the potential changes:
Sign Conventions
Batteries + if moving around the loop you pass from the − terminal to the + terminal. − if moving around the loop you pass from the + terminal to the − terminal.
Resistors
− if moving around the loop you are moving in the direction of the defined current (your arrows). + if you are moving against the defines current. Caution: If your circuit has multiple branches your loop path may go with the defined current in one branch and against it in another.

15. Example: Analyzing circuits with more that one loop+
Find the current and its direction through the 150 Ω resistor and the potential difference across it.
330 Ω
150 Ω
270 Ω
19 V
12 V

16. Example: Analyzing circuits with more that one loop
Draw the circuit and draw currents with an arrow in every separate branch of the circuit. (A branch is a section where the current does not change.)
Apply the junction law to enough junctions so that every current is used at least once.
junction eq. + I1
330 Ω
150 Ω
270 Ω I3
We could also have used this junction. We do not need to use both, because the information is redundant.
19 V
12 V I2

17. Example: Analyzing circuits with more that one loop
Apply the loop law to enough closed loops so that each current appears at least once. Remember the sign convention for the potential changes:
Sign Conventions
Batteries + if moving around the loop you pass from the − terminal to the + terminal. − if moving around the loop you pass from the + terminal to the − terminal.
Resistors
− if moving around the loop you are moving in the direction of the defined current (your arrows). + if you are moving against the defines current. Caution: If your circuit has multiple branches your loop path may go with the defined current in one branch and against it in another. + I1
330 Ω
150 Ω
270 Ω A I3
units check:
19 V
12 V I2
A Ω is the same as a V so we can add them.

18. Example: Analyzing circuits with more that one loop+ I1
loop equations
330 Ω
150 Ω
270 Ω A I3
19 V
12 V I2 + I1
330 Ω
150 Ω
270 Ω I3 B
19 V
12 V I2
The third loop is redundant and provides no new information. There are n−1 unique loops. Here, we can chose any two loop. +
330 Ω
150 Ω
270 Ω
19 V
12 V I2 I1 I3 C

19. Example: Analyzing circuits with more that one loop+
330 Ω
150 Ω
270 Ω
19 V
12 V I2 I1 I3
We have e equations and 3 unknowns.
Eliminate I1 first, let’s keep I3 because that is what we are asked to find.

20. Example: Analyzing circuits with more that one loop+
330 Ω
150 Ω
270 Ω
19 V
12 V I2 I1 I3

21. Example: Analyzing circuits with more that one loop+
330 Ω
150 Ω
270 Ω
19 V
12 V I2 I1 I3
Find the current and its direction through the 150 Ω resistor and the potential difference across it.

22. Clicker Question
Which is not a loop equation for the following circuit? + R1 E1 E2 R2 R3 IC IA IB A) B) C) D) E)

23. Ideal and Real Batteries
The ideal battery is a voltage source.
It maintains a constant voltage across its terminals.
It can supply any amount of current required to maintain its terminal voltage.
The current the battery supplies is determined by the rest of the circuit.
A real batteries have internal resistance.
Real batteries are simply an ideal voltage source (battery) with a series resistor representing its internal resistance.
The internal resistance limits the maximum current the battery can supply. + E + r E
ideal battery
real battery

24. Ammeters A A r Ammeter. An ammeter measures current.
It must be placed in series with the circuit in which the current is to be measured.
The ideal ammeter has zero internal resistance.
The ideal ammeter has zero voltage drop (the same as zero series resistance).
A real ammeter has internal resistance and is modeled as an ideal ammeter in series with a resistor, hence does contribute a voltage drop in the circuit it is measuring.
comments
The ammeter symbol is ALWAYS an ideal ammeter.
In problems always assume an ideal ammeter unless you are explicitly told otherwise. A r A
ideal ammeter
real ammeter

25. Voltmeters V V r Voltmeter.
A voltmeter measures voltage (potential difference).
It must be placed in parallel with the circuit in which the voltage is to be measured.
The ideal voltmeter has infinite internal resistance.
The ideal voltmeter has draws no current (the same as infinite resistance).
A real voltmeter has internal resistance and is modeled as an ideal voltmeter in parallel with a resistor, hence does draw current from the circuit it is measuring.
comments
The voltmeter symbol is ALWAYS an ideal voltmeter.
In problems always assume an ideal voltmeter unless you are explicitly told otherwise. V r V
ideal voltmeter
real voltmeter

26. Example: A real battery with EMF E is connected to an external resistor R. A voltmeter measures the potential difference across the resistor ΔVR. What is the internal resistance of the battery? What would the short circuit current of this battery be? E = 9.0 V; R = 17 Ω; ΔVR = 8.5 V

27. Clicker Question
What four identical resistor values R will have an equivalent resistance of 100 Ω when connected as shown? R A)
25 Ω
Req = 100 Ω B)
50 Ω C)
100 Ω D)
200 Ω E)
none of the above

28. Light Bulbs and Brightness
Electrical devices that emit light.
Incandescent lamps are resistors, that get hot enough to emit black body radiation.
The brightness of a lamp is proportional to the power dissipated in the lamp.
comments
The relative brightness of a light bulb in a circuit may be determined by finding the power dissipated in each bulb.
Well will ignore the changes in resistance with temperature of incandescent lamps. The filament in an incandescent lamp is typically made from tungsten because of its very high melting point. The resistivity of metals increases with increasing temperature. Because the filament operates at thousands of degrees above room temperature, the change in resistance from on to off can be very significant.
Light bulb

29. Light Bulb Brightness
The following circuit has 3 identical bulbs, a battery, and a switch.
Assumptions: The voltage across the battery will be constant. The resistances R of all three bulbs are the same.
Switch open:
When the switch is open, obviously bulb C is off.
We can simplify the circuit.
What can we say about the brightness of bulbs A and B?
Bulbs A & B are in series, thus the same current flows through each. Because the two bulbs also have the same resistance, the power (and therefore their brightness) will be the same.
What is the actual power in each bulb? I

30. Light Bulb Brightness Switch closed: We can simplify the circuit.
What can we say about the brightness of bulbs A, B, and C?
Bulbs B & C are in parallel, thus the same voltage across each. Because the two bulbs also have the same resistance, the power (and therefore their brightness) will be the same. IB IC IA
Bulb A is in series with the B & C. All the current flowing in the circuit passes through A. From Kirchhoff’s junction law we know that IA = IB + IC. Because the bulbs all have the same resistance, the current through A is twice that though B and C. Thus the power (and therefore their brightness) will be four times that of bulbs B and C.

31. Light Bulb Brightness Switch closed: We can simplify the circuit.
What can we say about the brightness of bulbs A, B, and C?
What is the actual power in each bulb? IB IC IA
Equivalent resistance of the circuit:
Power in bulb A.
Power in bulbs B and C.

32. Light Bulb Brightness
The following circuit has 3 identical bulbs, a battery, and a switch.
What can we say about the change in brightness of bulbs A and B when the switch is closed?
Qualitatively we can say:
The brightness of A increases because the current in the circuit increases as C is brought in parallel with B.
The brightness of B decreases because more voltage is dropped across A as the current in the circuit increases.
Quantitatively, we simply ratio the powers

33. Clicker Question
In the circuit to the right. The switch open. Compare the brightness of bulbs A and B. A)
A > B B)
A = B C)
A < B D)
There is no way to tell.

34. Clicker Question
In the circuit to the right. The switch open. Compare the brightness of bulbs B and C. A)
B > C B)
B = C C)
B < C D)
There is no way to tell.

35. Clicker Question
In the circuit to the right. When the switch is closed, the brightness of bulb B ______ . A)
decreases B)
increases C)
does not change D)
There is no way to tell.

36. Clicker Question
In the circuit to the right. The switch is now closed. Compare the brightness of bulbs B and C. A)
B > C B)
B = C C)
B < C D)
There is no way to tell.

37. Grounding
Grounding provides a convenient reference point for potential differences.
All potentials we measure are potential differences. However when we say a point in a circuit has a certain potential, we mean it has that potential with respect to a common reference point. Typically a ground connection defines that point.

38. Example: Adding a ground wire does not change the function of the circuit. It only changes the potentials with respect to ground. This is true when there is only ONE connection to ground so that no current can flow between the circuit and ground.
In this example, only the point of grounding is changed. Note that the currents and the potential differences with in the circuit are unchanged.

39. RC Circuits and Transients
So far we have considered capacitors that were charged and discharged. Now we will described how circuits with capacitors evolve from the charged state to the discharged state and vice versa.
The capacitor initially charged to Q0 = VC.
While the switch is open no current flows in the circuit.
The switch is closed at t = 0 current begins to flow and the capacitor begins to discharge.
I0 = V/R
We can use Kirchhoff’s Loop Law C R
I = 0
before switch is closed
ΔVR = 0
ΔVC = Q0/C
ΔVC = Q/C C R
ΔVR = IR I
after switch is closed
τ is the time constant, the rate the capacitor charges and discharges
units:

40. RC Circuits and Transients
The time constant τ is determined by R and C only. It is the rate the circuit evolves from the initial state (charged) to the final state (discharged).
The exponential decay of the charge decreases by a factor e−1 over each time interval τ.
The current also decays exponentially with time.

41. RC Circuits and Transients
The charging process also follows exponential behavior with time constant τ.
The capacitor initially discharged with Q0 = 0.
While the switch is open no current flows in the circuit.
The switch is closed at t = 0.
I0 = V/R
The capacitor charges to Qmax following a complimentary exponential.
The exponential decay of the charge difference (Qmax − Q) decreases by a factor e−1 for each time interval τ.

42. Example: The switch has been in position a for a long time
Example: The switch has been in position a for a long time. It is changed to position b at t = 0. What is the time constant of the circuit? What is the charge on the capacitor and the current through the circuit after time t = 4.7 μs? How long does it take for the capacitor to loose half its charge?
E = 9.0 V; R = 17 Ω; C = 1.5 μF E C R

43. Example: The switch has been in position a for a long time
Example: The switch has been in position a for a long time. It is changed to position b at t = 0. What is the time constant of the circuit? What is the charge on the capacitor and the current through the circuit after time t = 4.7 μs? How long does it take for the capacitor to loose half its charge?
E = 9.0 V; R = 17 Ω; C = 1.5 μF E C R

44. Clicker Question
Which of the following are not units of RC (resistance × capacitance)? A) B) C) D) E)

46. Stop here

48. Ammeters and Voltmeters
An ammeter measures current.
It must be placed in series with the circuit in which the current is to be measured.
The ideal ammeter has zero internal resistance.
Voltmeter.
A voltmeter measures voltage (potential difference).
It must be placed in parallel with the circuit in which the voltage is to be measured.
The ideal voltmeter has infinite internal resistance. A V
ideal ammeter
ideal voltmeter