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The following problems and their diagrams were written by Mr. Donald Kieffer. The guided solutions to these problems have been written by myself.

ELECTRICAL CIRCUITS

SERIES CIRCUITS

If several electrical devices are connected end to end in a circuit, electricity must flow through each in succession. There is a single path for the moving charges, and so the same current must be in each device. Otherwise, there would be an accumulation of charge at different points around the conducting circuit. We know that a charge can be accumulated on a conductor only if it is isolated. An electric circuit with components arranged to provide a single conducting path for electrical charges is known as a SERIES CIRCUIT.

There are several inherent characteristics with series circuit operations. If one element of a series connection fails to provide a conducting path, the circuit is said to be open. Also, each component of the circuit offers opposition to the flow of electricity, and resistance in a circuit limits the magnitude of the electron flow in the circuit according to OHMS LAW. When connected in series, resistance is cumulative; the more resistance components that you have in the circuit, the higher the total resistance (opposition to flow of electrons), and the smaller the current (electron flow) in the circuit for a given applied EMF (voltage). Obviously, since there must be a single magnitude of current in the circuit, all devices connected in series with each other must be designed to function at the same current magnitude.

From Ohm’s LAW, the drop in potential (voltage) across each component of a series circuit is the product of the current flowing through the circuit and the resistance (opposition to electron flow) of the components. Electrons lose energy as they go (fall) through a difference in potential; in a series circuit these losses occur in succession and are therefore cumulative. The drops in potential (voltage) across successive components in a series circuit are additive and their sum is equal then to the total potential (voltage) difference across the entire series circuit or series branch.

We may summarize these observations by stating three CARDINAL RULES for circuit elements that are connected in SERIES:

1. The current in all parts of a series circuit is the same

I_T = I₁ = I₂ = I₃ = . . . = I_N

2. The sum of all the individual drops (losses) in potential around a series circuit is equal

to the total applied potential (voltage) to that circuit or branch

V_T = V₁ + V₂ + V₃ + . . . Vn = ΣV_N

3. The total resistance in a series circuit is equal to the sum of all the separate

resistance in that circuit or branch

R_T = R₁ + R₂ + R₃ + . . . Rn = ΣR_N

Suppose the above rules are applied to a circuit consisting of several elements as shown in the accompanying diagram. The computations are shown with the circuit diagram. Observe that the positive side of “R1” is 3.0 volts positive with respect to the cathode of the battery and is 9 volts negative with respect to the anode of the battery. The three potential (drops) losses in the circuit are in series and therefore their sum equals the magnitude of the total applied EMF (voltage), but their signs are opposite.

Stated in another way, the algebraic sum of all the changes in potential (voltage) occurring around the completed circuit is equal to zero.

R_T = R₁ + R₂ + R₃ = 6 ohms + 8 ohms + 10 ohms = 24 ohms

I = E/R_T = 12 volts/24 ohms = 0.50 amps

E_T = E₁ + E₂ + E₃ = (I)(R₁) + (I)(R₂) + (I)(E₃)

E_T = 3.0 v + 4.0 v + 5.0 v = 12.0 v

Problems to Solve

Find: V_T; R_T; V₁; V₂; V₃; V₄

FIND: V_T; R_T; I_T; V₁; V₂; V₃; V₄

FIND: R_T; R₆; V₁; V₂; V₃; V₄; V₅; V₆

FIND: R_T; I_T; V₁; V₂; V₃; V₄; V₅; V₆

FIND: R₁; R₅; R_T; V_T; V₂; V₃; V₄

FIND: V_T; R_T; I_T; V₁; V₂; V₃; V₄; V₅; V₆

FIND: R_T; V_T; I_T; V₁; V₂; V₃; V₄; V₅; V₆; V₇

PARALLEL CIRCUITS

Most electrical circuits you will encounter in practice are parallel circuits. House and apartment wiring circuits consist of a multiplicity of parallel connections and current paths, all fed by the same source of potential (voltage). The parallel circuit is basically different from the series type, in that the current divides into a number of separate, independent paths or branches. Each of these branches or paths may have a different resistance (opposition to the flow of electrons) and, therefore, the value of the current (actual electron flow) in each path or branch may be different. If one of the elements burns out or is disconnected, the remainder of the circuit continues to operate. This is a great advantage over the series circuit, where if one element burns out the electrons will stop flowing in all parts of the circuit. Furthermore, since all paths or branches operate on the same potential (voltage) source, one single source of the proper potential and power rating is able to supply the currents to all the parallel branches. The check for a parallel circuit is simple: If you trace MORE THAN ONE PATH OR BRANCH FOR THE CURRENT FLOW, then you have a PARALLEL CIRCUIT.

The chief advantage of the parallel circuit is that when one path or branch of the circuit is opened (broken), the current will still continue to flow through the rest of the circuit. However, one draw back to a parallel circuit is that as more and more branches are placed in parallel with each other, the effective resistance (opposition to flow of electrons) decreases which in turn causes an increase in electron flow (amps). Also a parallel circuit acts as a current divider whereas the series circuit acts as a voltage divider.

By Ohm’s Law, the voltage loss (drop) across (through) each branch element is the product of the current (electron flow) and the opposition (resistance) that branch or path provides, and this loss in potential (voltage) must equal the total applied potential (voltage) if it is a pure parallel circuit. Also from Ohm’s Law, the current is equal to the potential (voltage) divided by the resistance (opposition to flow). [That is the total current in the branch is equal to the potential (voltage) for that branch divided by the total opposition (resistance) in that branch.] Now mathematically, if the voltage (potential) remains the same in parallel circuits (pure) and the current (electron flow) has the opportunity to fluctuate due to the opposition that is located in the individual branches, then the total opposition of the parallel branches or paths effectively must decrease according to Ohm’s Law.

Let us summarize then the observations concerning parallel circuits by stating THREE CARDINAL RULES for PARALLEL CIRCUITS:

1. The total current in a parallel circuit is equal to the sum of the currents in the separate

branches or paths.

I_T = I₁ + I₂ + I₃ + . . . I_N = Σ I_N

2. The potential difference across all branches of a parallel circuit must have the same

magnitude.

V_T = V₁ = V₂ = V₃ = . . . V_N = Σ V_N

3. The total opposition of flow of electrons, since it must decrease is equal to the sum of the

reciprocals. That is the reciprocal of the total resistance is equal to the sum of the

reciprocals of the separate branches or paths that are in parallel with each other.

(1/R_T) = (1/R₁) + (1/R₂) + (1/R₃) + . . . (1/R_N) or (1/R_T) = Σ (1/R_N)

Suppose these rules for parallel circuits are now applied to a circuit consisting of several resistors as show in the accompanying diagram. The computations a re shown with the diagram and the polarity of the elements are indicated and are consistent with the anode and cathode of the source of potential. Ammeters and voltmeters are inserted into the diagram to illustrate the concepts previously stated.

I₁ = (V_T)/(R₁) = (30 v)/(5 ohm) = 6.0 amp

I₂ = (V_T)/(R₂) = (30 v)/(10 ohm) = 3.0 amp

I₃ = (V_T)/(R₃) = (30 v)/(15 ohm) = 2.0 amp

I₁ + I₂ + I₃ = I_T = 11.0 amp

(1/R_T) = (1/R₁) + (1/R₂) + (1/R₃) = (1/5 ohm) + (1/10 ohm) + (1/15 ohm); RT = 2.727 ohms

Check

I_T = (V_T)/(R_T) = (30 v)/(2.727 ohms) = 11.0 amp

PROBLEMS TO SOLVE

FIND: V_T; R_T; A₁; A₂; A₃; A₄; A₅

FIND: V_T; R_T; A₁; A₂; A₄; A₅

FIND: A₁; A₃; A₄; R_T; V_T

FIND: A₁; A₂; A₃; A₄; A₅; R_T

FIND: V_T; R_T; A₂; A₃; A4

7.Ten lamps are connected in parallel with a source of potential which furnishes 110 v. (A) What

is the potential loss across each lamp? (B) What is the current flow in each lamp? (C) What is

the total resistance in the circuit?

8. The lamps in question # 7 are now connected together in a series configuration. (A) What is

the potential loss across each lamp? (B) What current flows through each lamp? (C) What is

the total resistance of the circuit?

9. An electric heater operating on 120 v has ten wires of equal resistance connected in parallel

with each other. The resistance of each wire is 40 ohms. (A) What current flows through each

wire? (B) What total current passes through the heater as a whole?

10. Three cells, each having an internal resistance of 0.05 ohms and a source of potential of

1.5 volt, are connected in series with four resistances of 16 ohms each. (A) What current will

flow in this circuit? (B) What current will flow through each part of this circuit? (C) What is the

total resistance of the circuit?

11. The three cells in problem # 10 are now connected in a parallel configuration. (A) What

current will flow in the circuit? (B) What current will flow through each part? (C) What is the

total resistance of the circuit?

12. The cells in problem # 10 are connected in a series configuration again, but the resistors are

connected in parallel. (A) What current will flow in the circuit? (B) What current will flow

through each part? (C) What is the total resistance of the circuit?

13. A network of resistors is connected to a 440 v source of potential. (A) What is the effective

resistance of a series network made up of 75 ohm resistors and one 3 ohm resistor? (B) What

is the current flow through each?

14. What would be the power consumption in watts of each resistor in problem # 13?

15. What would be the power consumption of the entire circuit in problem # 8?

16. What would be the power consumption of the entire circuit in problem # 10?

17. (A) How does the power consumption of the entire circuit in problem # 11 compare to that in

problem # 10? (B) How does the power consumption of each resistor in problem # 11

compare to that in problem # 10? (C) Why would it differ, if it does?

18. (A) How does the power consumption of the entire circuit in problem # 12 compare to that in

problem # 10 and # 11? (B) How does the power consumption of each resistor in problem

# 12 compare to that in problem # 10 and # 11? (C) Why would it differ, if it does?

19. Your house is 400 ft from the electric power line which delivers a potential of 120 v. You

have a special pair of feeder lines whose total resistance is 0.20 ohms. Due to an equipment

failure outside your house, the feeder lines arc and draw 180 amp. (A) What is the potential

at your house when this occurs? (B) What is the potential at your house when you turn on

switches to appliances that draw an additional 20 amp?

20. A 150 watt lamp consumes 150 watts of power and converts it into heat and light when it

operates on a 120 v source of potential. (A) How much current does this lamp pull from the

source of potential? (B) What is the resistance of the lamp at this time?

21. An clothes iron operating on 120 v uses 8 amp of current has a connecting wire whose

resistance is 0.20 ohms. (A) What power is wasted in the connecting wire? (B) What is the

resistance of the clothes iron by itself? (C) What power does the clothes iron itself consume?

22. A 500 ohm resistor in a circuit has a potential of 5 v across its terminals. What power

appears as heat in this resistor?

23. (A) What current is involved in the normal operation of a 120 v lamp rated at: (1) 50 watts?

(2) 150 watts? (3) 1 kwatt? (B) Does a resistor with a higher wattage rating necessarily have

a higher resistance? Why?

24. In a laboratory experiment using 60 watt light bulbs, the potential applied was varied as

follows: (A) 30 v; (B) 50 v; (C) 90 v; (D) 120 v; (E) 180 v; and (F) 200v. (A) What was the

current draw during each of the trials? (B) What was the resistance of the lamp during each

one of the trials?

SELF-TEST

Completion - Use the term that BEST completes the statement.

1. Electric current transfers from one place to another.

2. When electrons are forced away from a(n) plate, work is done.

3. The difference in potential between any two points in a field is the required to move a unit of whatever is affected by the field between two points.

4. is the unit for electric field intensity.

5. Electrons flow from the plate to the plate in a battery.

6. generators are devices that borrow electrons from one object and place them on another object to “generate” a charge.

7. In electric terminology, one joule/coulomb is equivalent to one .

8. The unit of electric is a joule/sec or watt.

9. Current flow of one coulomb/sec is called 1 .

10. discovered that the ratio of the potential difference between the end of a conductor and the current flowing through that conductor is a constant.

11. Electrical current flowing through a conductor varies to the potential applied to that conductor.

12. The unit of electrical resistance is called the .

13. A should always be connected in parallel across a resistance and it measures the in the circuit.

14. A should always be placed in line or in series in a circuit and it measures the in the circuit.

15. A should always be placed in parallel or across a resistance without the of ____ connected and it measures the of that circuit/element.

16. The energy consumed by an element(s) in a circuit is called and it is directly proportional tothe product of the resistance and the square of the through the resistor, or the product of and .

Multiple Choice

1. Which of the following is NOT the function of a battery/cell/generator in an electric circuit?

(A) to supply energy to drive electrons through the circuit.

(B) to maintain a constant difference of potential between two points in the circuit.

(D) to convert chemical energy to electrical energy.

2. The EMF of a source of potential may be expressed as volts or:

(A) newtons; (C) joules/coulomb;

(B) joules; (D) watts.

3. In a closed circuit, the EMF of a source of potential is greater than the terminal voltage because:

(A) the source has chemical defects;

(B) some of the energy furnished is used to drive the current through the source of potential itself.

(D) the resistance of the external circuit opposes the flow of electrons through it.

4. Which of the following pairs of units BEST describes first a unit of electrical power and second a

unit of work:

(A) W, (W)(s); (C) (W)/(s); J;

(B) V; J/s; (D) V; (kW)(h)

5. A difference in potential of 30 volts applied to a resistance maintains a current of 2.0 amperes for

10 seconds. The power supplied to the resistance is:

(A) 60 W; (C) 6 W;

(B) 150 W; (D) 600 W.

6. The electrical energy supplied to the circuit in question #5 is:

(A) 60 W; (C) 6(W)(s);

(B) 600 J; (D) 20 (A)(s)

7. The electrical energy supplied to a 100 W lamp in 2.0 s is:

(A) 50 J; (C) 12,000 J;

(B) 200 J; (D) 5/6 J.

8. A 24 W lamp is operated by a 6 V battery. The current flowing through the lamp is:

(A) 1/4 A; (C) 18 A;

(B) 4 A; (D) 30 A.

9. If the lamp in question #8 has an incandescent filament, as the applied voltage increases the

current will:

(A) increase proportionally; (C) increase, but not proportionally;

(B) decrease proportionally; (D) decrease, but not proportionally.

10. A metal conductor obeys Ohm’s Law only if:

(A) it is kept at constant temperature: (C) the current flowing through it is large;

(B) its resistance is very small; (D) the difference of potential applied to it is varied.

11. A 12 V source of potential provides a steady current of 2.0 amperes through an iron wire. The

resistance of the wire is:

(A) 24 Ohm; (C) 10 Volt;

(B) 14 Ohm; (D) 6 Ohm.

12. The wire in question #11 is cooled and maintained at a temperature of -100^o C, the resistance of the

wire will:

(A) drop to zero; (C) increase;

(B) decrease; (D) remain the same.

13. Equal currents flow through a 2 Ohm and a 3 Ohm resistor. The ratio of the rate at which the 3 Ohm

resistor generates heat to the rate at which the 2 Ohm resistor generates heat is:

(A) 3 to 2; (C) 0.9 to 4;

(B) 2 to 3; (D) 4 to 9

14. If equal currents were to flow through the two resistors in question #13, the resistors would have to

connected together in what type of configuration:

(A) series;

(B) parallel.

15. Two resistors are connected together such that they both have the same potential applied. They

must be configured how:

(A) series;

(B) parallel.

16. When two different resistors are connected in series, both resistors:

(A) have the same potential difference across them; (C) generate equal quantities of heat;

(B) have the same current; (D) consume equal quantities of power.

17. When two different resistors are connected in parallel, both resistors:

(A) have the same potential difference across them; (C) generate equal quantities of heat;

(B) have the same current; (D) consume equal quantities of power.

18. The combined resistance of a 5 Ohm and a 10 Ohm resistor connected in parallel to the same

source of potential is:

(A) 2 Ohm; (C) 7 1/2 Ohm;

(B) 3 1/3 Ohm; (D) 15 Ohm.

19. The combined resistance of a 5 Ohm and a 10 Ohm resistor connected in series to the same

source of potential is:

(A) 2 Ohm; (C) 7 1/2 Ohm;

(B) 3 1/3 Ohm; (D) 15 Ohm.

20. Several 440 Watt heaters are connected in parallel to a 110-Volt source of potential. Each heater

may be turned on and off individually. The fuse in circuit will only allow a maximum of 15 amperes

too flow in the circuit. The maximum number of heaters that can be simultaneously operated without

blowing the fuse is:

(A) 1; (D) 4;

(B) 2; (E) 5;

21. Several 440 Watt heaters are connected in series to each other and to a 110-Volt source of

potential. Each heater may be turned on and off individually. The fuse in circuit will only allow

a maximum of 15 amperes to flow in the circuit. The maximum number of heaters that can be

simultaneously operated without blowing the fuse is:

(A) 1; (D) 4;

(B) 2; (E) 5;

Use the following diagram to answer questions # 22 through # 28.

22. As the source of potential is doubled, what happens to the current through R₁:

(A) increases;

(B) decreases;

23. As the value of resistor R₂ increases, the potential loss across R₃:

(A) increases;

(B) decreases;

24. As the value of resistor R₃ increases, what happens to the total circuit resistance:

(A) increases;

(B) decreases;

25. Where would you place an Ammeter in the above circuit to obtain the largest reading?

What would it read? _________

26. Where would you place an Voltmeter in the above circuit to obtain the largest reading?

What would it read? _________

27. If a 1 Ohm resistor were placed in parallel to the above circuit; what happens to the total current

flowing in the circuit?

(A) increases;

(B) decreases;

28. What would happen to the Ammeter reading in question # 25 when the resistor in question # 27 is

placed in the circuit? What would it read? _________

29. What would happen to the Voltmeter reading in question # 26 when the resistor in question # 27 is

placed in the circuit? What would it read? .

Use the following diagram for questions # 30 through # 38

30. As the source of potential is doubled, what happens to the current through R₁:

(A) increases;

(B) decreases;

31. As the value of resistor R₂ increases, the potential loss across R₃:

(A) increases;

(B) decreases;

32. As the value of resistor R₃ increases, what happens to the total circuit resistance:

(A) increases;

(B) decreases;

33. If a 1 Ohm resistor were placed in series to the above circuit; what happens to the total current

flowing in the circuit?

(A) increases;

(B) decreases;

34. Where would you place an Ammeter in the above circuit to obtain the largest reading?

What would it read? _________

35. Where would you place an Voltmeter in the above circuit to obtain the largest reading?

What would it read? _________

36. If a 1 Ohm resistor were placed in parallel to the above circuit; what happens to the total current

flowing in the circuit?

(A) increases;

(B) decreases;

37. What would happen to the Ammeter reading in question # 34 when the resistor in question

# 36 is placed in the circuit? What would it read? .

38. What would happen to the Voltmeter reading in question # 35 when the resistor in question

# 36 is placed in the circuit? What would it read? .

SERIES-PARALLEL COMBINATION CIRCUITS

FIND: R_T; A₁; A₂; A₃; A₄; Voltage drop across each resistor

FIND: R_T; A₁; A₂; A₃; V₁; V₂; V₃

FIND: V_T; R_T; A₁; A₂; A₃; A₄; A₅; A₆

FIND: A₁; A₂; A₃; A₄; A₅; R_T; potential loss (V) across each resistor

FIND: A₁; A₂; A₃; A₄; A₅; A₆; V₁; V₂; R_T; potential loss (V) across each resistor

FIND: A₁; A₂; A₃; R_T; potential loss (V) across each resistor

FIND: R_T; A₁; A₂; A₃; A₄; potential loss (V) across each resistor

FIND: R_T; A₁; A₂; A₃; potential loss (V) across each resistor

10.

FIND: R_T; A₁; A₂; A₃; potential loss (V) across each resistor

11.

FIND: R_T; A₁; A₂; A₃; potential loss (V) across each resistor

12.

FIND A₁; A₂; A₃; A₄; A₅; R_T; potential loss (V) across each resistor

13.

FIND: A₁; A₂; A₃; A₄; A₅; R_T; potential loss (V) across each resistor

14.

FIND A₁; A₂; A₃; A₄; A₅; R_T; potential loss (V) across each resistor

15.

FIND: A₁; A₂; A₃; A₄; A₅; R_T; potential loss (V) across each resistor

16.

FIND: A₁; A₂; A₃; A₄; A₅; R_T; potential loss (V) across each resistor

17.

FIND: A₁; A₂; A₃; A₄; A₅; R_T; potential loss (V) across each resistor; and

voltmeter readings 1 thru 3

SELF-TEST

Use the following diagram to answer questions #1 through #3

1. As the source of potential is doubled, what happens to the current through R₁?

(A) Increases;

(B) decreases;

2. The value of resistor R₂ increases, the potential loss across R₃:

(A) increases;

(B) decreases;

3. As the value of resistor R₃ increases, what happens to the total circuit resistance:

(A) increases;

(B) decreases;

4. Where would you place an ammeter in the above circuit to obtain the largest reading? ____________.

It would read? __________.

5. Where would you place a voltmeter in the above circuit to obtain the largest reading? ___________.

It would read? ___________.

6. If a 10 ohm resistor were placed in parallel to the above circuit, what would happen to the total current

flowing in the circuit (as compared to the original):

(A) increase; Its value would be? ________

(B) decrease; Its value would be? ________

7. What would happen to the ammeter reading in question #4 when the above mentioned 10 ohm

resistor is in place as in question #6?

(A) increase; Its value would be? ________

(B) decrease; Its value would be? ________

8. What would happen to the voltmeter reading in question #5 when the above mentioned 10 ohm

resistor is in place as in question #6?

(A) increase; Its value would be? ________

(B) decrease; Its value would be? ________