Content check — College Physics (AP), Chapter 21: Circuits and DC Instruments

Note: Data taken from figures can be assumed to be accurate to three significant digits.

(a) What is the resistance of ten $275-Ω$ resistors connected in series? (b) In parallel?

Answer

(a) $2.75 k Ω$

(b) $27.5 Ω$

Note: Data taken from figures can be assumed to be accurate to three significant digits.

What are the largest and smallest resistances you can obtain by connecting a $36 . 0-Ω$ , a $50 . 0-Ω$ , and a $700-Ω$ resistor together?

Answer

(a) $786 Ω$

(b) $20.3 Ω$

21-5 §21.1 source ↗

Note: Data taken from figures can be assumed to be accurate to three significant digits.

Your car’s 30.0-W headlight and 2.40-kW starter are ordinarily connected in parallel in a 12.0-V system. What power would one headlight and the starter consume if connected in series to a 12.0-V battery? (Neglect any other resistance in the circuit and any change in resistance in the two devices.)

Answer

29.6 W

21-7 §21.1 source ↗

Note: Data taken from figures can be assumed to be accurate to three significant digits.

Referring to the example combining series and parallel circuits and Figure 21.6, calculate $I_{3}$ in the following two different ways: (a) from the known values of $I$ and $I_{2}$ ; (b) using Ohm’s law for $R_{3}$ . In both parts explicitly show how you follow the steps in the Problem-Solving Strategies for Series and Parallel Resistors.

Circuit diagram in which a battery of twelve point zero volts is connected to a combination of three resistors. Resistors R sub two and R sub three are connected in parallel to each other, and their combination is connected in series to resistor R sub one. R sub one has a resistance of one point zero zero ohms, R sub two has a resistance of six point zero zero ohms, and R sub three has a resistance of thirteen point zero ohms. — Figure 21.6 in the book ↗

Answer

(a) 0.74 A

(b) 0.742 A

21-9 §21.1 source ↗

Note: Data taken from figures can be assumed to be accurate to three significant digits.

Refer to Figure 21.7 and the discussion of lights dimming when a heavy appliance comes on. (a) Given the voltage source is 120 V, the wire resistance is $0.400 Ω$ , and the bulb is nominally 75.0 W, what power will the bulb dissipate if a total of 15.0 A passes through the wires when the motor comes on? Assume negligible change in bulb resistance. (b) What power is consumed by the motor?

A conceptual drawing showing a refrigerator with its motor and light bulbs connected to a household A C circuit through a wire with resistance of R sub one. The bulb has a resistance R sub two, and the motor has a resistance R sub three. — Figure 21.7 in the book ↗

Answer

(a) 67.7 W

(b) 1.64 kW

21-11 §21.1 source ↗

Note: Data taken from figures can be assumed to be accurate to three significant digits.

Show that if two resistors $R_{1}$ and $R_{2}$ are combined and one is much greater than the other ( $R_{1} >> R_{2}$ ): (a) Their series resistance is very nearly equal to the greater resistance $R_{1}$ . (b) Their parallel resistance is very nearly equal to smaller resistance $R_{2}$ .

Answer

(a) $\begin{matrix} R_{s} = R_{1} + R_{2} \\ \Rightarrow R_{s} \approx R_{1} R_{1} >> R_{2} \end{matrix}$

(b) $\frac{1}{R_{p}} =$ $\frac{1}{R_{1}} +$ $\frac{1}{R_{2}} =$ $\frac{R_{1} + R_{2}}{R_{1} R_{2}}$ ,

so that

$\begin{matrix} R_{p} = \frac{R_{1} R_{2}}{R_{1} + R_{2}} \approx \frac{R_{1} R_{2}}{R_{1}} = R_{2} R_{1} >> R_{2} . \end{matrix}$

21-13 §21.1 source ↗

Note: Data taken from figures can be assumed to be accurate to three significant digits.

Unreasonable Results

Two resistors, one having a resistance of $9 00 kΩ$ , are connected in series to produce a total resistance of $0 . 500 MΩ$ . (a) What is the value of the second resistance? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Answer

(a) $- 400 k Ω$

(b) Resistance cannot be negative.

(c) Series resistance is said to be less than one of the resistors, but it must be greater than any of the resistors.

21-14 §21.2 source ↗

Standard automobile batteries have six lead-acid cells in series, creating a total emf of 12.0 V. What is the emf of an individual lead-acid cell?

Answer

2.00 V

21-16 §21.2 source ↗

What is the output voltage of a 3.0000-V lithium cell in a digital wristwatch that draws 0.300 mA, if the cell’s internal resistance is

2.00 Ω

?

Answer

2.9994 V

21-18 §21.2 source ↗

What is the internal resistance of an automobile battery that has an emf of 12.0 V and a terminal voltage of 15.0 V while a current of 8.00 A is charging it?

Answer

0.375 Ω

21-21 §21.2 source ↗

The hot resistance of a flashlight bulb is

2.30 Ω

, and it is run by a 1.58-V alkaline cell having a

0 . 100-Ω

internal resistance. (a) What current flows? (b) Calculate the power supplied to the bulb using

I^{2} R_{bulb}

. (c) Is this power the same as calculated using

\frac{V^{2}}{R_{bulb}}

?

Answer

(a) 0.658 A

(b) 0.997 W

(c) 0.997 W; yes

21-23 §21.2 source ↗

An automobile starter motor has an equivalent resistance of

0.0500 Ω

and is supplied by a 12.0-V battery with a

0 . 0100-Ω

internal resistance. (a) What is the current to the motor? (b) What voltage is applied to it? (c) What power is supplied to the motor? (d) Repeat these calculations for when the battery connections are corroded and add

0.0900 Ω

to the circuit. (Significant problems are caused by even small amounts of unwanted resistance in low-voltage, high-current applications.)

Answer

(a) 200 A

(b) 10.0 V

(c) 2.00 kW

(d) $0.1000 Ω; 80 . 0 A, 4 . 0 V, 320 W$

21-25 §21.2 source ↗

(a) What is the internal resistance of a voltage source if its terminal voltage drops by 2.00 V when the current supplied increases by 5.00 A? (b) Can the emf of the voltage source be found with the information supplied?

Answer

(a) $0 . 400 Ω$

(b) No, there is only one independent equation, so only $r$ can be found.

21-29 §21.2 source ↗

Unreasonable Results

A 1.58-V alkaline cell with a $0 . 200-Ω$ internal resistance is supplying 8.50 A to a load. (a) What is its terminal voltage? (b) What is the value of the load resistance? (c) What is unreasonable about these results? (d) Which assumptions are unreasonable or inconsistent?

Answer

(a) –0.120 V

(b) $- 1 . 41 \times 10^{- 2} Ω$

(c) Negative terminal voltage; negative load resistance.

(d) The assumption that such a cell could provide 8.50 A is inconsistent with its internal resistance.

21-31 §21.3 source ↗

Apply the loop rule to loop abcdefgha in Figure 21.25.

The diagram shows a complex circuit with two voltage sources E sub one and E sub two and several resistive loads, wired in two loops and two junctions. Several points on the diagram are marked with letters a through h. The current in each branch is labeled separately. — Figure 21.25 in the book ↗

Answer

- I_{2} R_{2} + E_{1} - I_{2} r_{1} + I_{3} R_{3} + I_{3} r_{2} - E_{2} = 0

21-35 §21.3 source ↗

Apply the junction rule at point a in Figure 21.50.

The diagram shows a complex circuit with four voltage sources E sub one, E sub two, E sub three, E sub four and several resistive loads, wired in two loops and many junctions. Several points on the diagram are marked with letters a through j. The current in each branch is labeled separately. — Figure 21.50 in the book ↗

Answer

I_{3} = I_{1} + I_{2}

21-37 §21.3 source ↗

Apply the loop rule to loop akledcba in Figure 21.50.

Answer

{emf}_{2} - I_{2} r_{2} - I_{2} R_{2} + I_{1} R_{5} + I_{1} r_{1} - {emf}_{1} + I_{1} R_{1} = 0

21-39 §21.3 source ↗

Solve Example 21.5, but use loop abcdefgha instead of loop akledcba. Explicitly show how you follow the steps in the Problem-Solving Strategies for Series and Parallel Resistors.

Example 21.5 Calculating Current: Using Kirchhoff’s Rules in the book ↗

Answer

(a) $I_{1} = 4.75 A$

(b) $I_{2} = - 3 . 5 A$

(c) $I_{3} = 8 . 25 A$

21-41 §21.3 source ↗

Unreasonable Results

Consider the circuit in Figure 21.51, and suppose that the emfs are unknown and the currents are given to be $I_{1} = 5 . 00 A$ , $I_{2} = 3 .0 A$ , and $I_{3} = –2 . 00 A$ . (a) Could you find the emfs? (b) What is wrong with the assumptions?

The diagram shows a complex circuit with two voltage sources E sub one and E sub two, and three resistive loads, wired in two loops and two junctions. Several points on the diagram are marked with letters a through h. The current in each branch is labeled separately. — Figure 21.51 in the book ↗

Answer

(a) No, you would get inconsistent equations to solve.

(b) $I_{1} \neq I_{2} + I_{3}$ . The assumed currents violate the junction rule.

21-42 §21.4 source ↗

What is the sensitivity of the galvanometer (that is, what current gives a full-scale deflection) inside a voltmeter that has a

1.00 -M Ω

resistance on its 30.0-V scale?

Answer

30 μA

21-44 §21.4 source ↗

Find the resistance that must be placed in series with a

25 . 0-Ω

galvanometer having a

50.0 -μA

sensitivity (the same as the one discussed in the text) to allow it to be used as a voltmeter with a 0.100-V full-scale reading.

Answer

1 . 98 k Ω

21-46 §21.4 source ↗

Find the resistance that must be placed in parallel with a

25 . 0-Ω

galvanometer having a

50 . 0-μA

sensitivity (the same as the one discussed in the text) to allow it to be used as an ammeter with a 10.0-A full-scale reading. Include a circuit diagram with your solution.

Answer

1.25 \times 10^{- 4} Ω

21-48 §21.4 source ↗

Find the resistance that must be placed in series with a

10 . 0-Ω

galvanometer having a

100-μA

sensitivity to allow it to be used as a voltmeter with: (a) a 300-V full-scale reading, and (b) a 0.300-V full-scale reading.

Answer

(a) $3 . 00 M Ω$

(b) $2 . 99 k Ω$

21-50 §21.4 source ↗

Suppose you measure the terminal voltage of a 1.585-V alkaline cell having an internal resistance of

0.100 Ω

by placing a

1.00 -k Ω

voltmeter across its terminals. (See Figure 21.52.) (a) What current flows? (b) Find the terminal voltage. (c) To see how close the measured terminal voltage is to the emf, calculate their ratio.

The figure shows a circuit diagram that includes a battery with an internal resistance r and a voltmeter connected across its terminals. The current I is shown by an arrow pointing in a clockwise direction. — Figure 21.52 in the book ↗

Answer

(a) 1.58 mA

(b) 1.5848 V (need four digits to see the difference)

(c) 0.99990 (need five digits to see the difference from unity)

21-52 §21.4 source ↗

A certain ammeter has a resistance of

5.00 \times 10^{- 5} Ω

on its 3.00-A scale and contains a

10 . 0-Ω

galvanometer. What is the sensitivity of the galvanometer?

Answer

15 . 0 μA

21-54 §21.4 source ↗

A

0 . 0200-Ω

ammeter is placed in series with a

10 . 00-Ω

resistor in a circuit. (a) Draw a circuit diagram of the connection. (b) Calculate the resistance of the combination. (c) If the voltage is kept the same across the combination as it was through the

10 . 00-Ω

resistor alone, what is the percent decrease in current? (d) If the current is kept the same through the combination as it was through the

10 . 00-Ω

resistor alone, what is the percent increase in voltage? (e) Are the changes found in parts (c) and (d) significant? Discuss.

Answer

(a)

(b) $10.02 Ω$

(c) 0.9980, or a $2.0 \times 10^{–1}$ percent decrease

(d) 1.002, or a $2.0 \times 10^{–1}$ percent increase

(e) Not significant.

21-56 §21.4 source ↗

Unreasonable Results

(a) What resistance would you put in parallel with a $40 . 0-Ω$ galvanometer having a $25. 0-μA$ sensitivity to allow it to be used as an ammeter that has a full-scale deflection for $10 . 0-μA$ ? (b) What is unreasonable about this result? (c) Which assumptions are responsible?

Answer

(a) $- 66 . 7 Ω$

(b) You can’t have negative resistance.

(c) It is unreasonable that $I_{G}$ is greater than $I_{tot}$ (see Figure 21.30). You cannot achieve a full-scale deflection using a current less than the sensitivity of the galvanometer.

A resistance R is placed in parallel with a galvanometer G having an internal resistance r to produce an ammeter. — Figure 21.30 in the book ↗

21-57 §21.5 source ↗

What is the

{emf}_{x}

of a cell being measured in a potentiometer, if the standard cell’s emf is 12.0 V and the potentiometer balances for

R_{x} = 5 . 000 Ω

and

R_{s} = 2 . 500 Ω

?

Answer

24.0 V

21-59 §21.5 source ↗

When an unknown resistance

R_{x}

is placed in a Wheatstone bridge, it is possible to balance the bridge by adjusting

R_{3}

to be

2500 Ω

. What is

R_{x}

if

\frac{R_{2}}{R_{1}} = 0 . 625

?

Answer

1 . 56 k Ω

21-61 §21.5 source ↗

(a) What is the unknown

{emf}_{x}

in a potentiometer that balances when

R_{x}

is

10.0 Ω

, and balances when

R_{s}

is

15.0 Ω

for a standard 3.000-V emf? (b) The same

{emf}_{x}

is placed in the same potentiometer, which now balances when

R_{s}

is

15.0 Ω

for a standard emf of 3.100 V. At what resistance

R_{x}

will the potentiometer balance?

Answer

(a) 2.00 V

(b) $9.68 Ω$

21-62 §21.5 source ↗

Suppose you want to measure resistances in the range from

10.0 Ω

to

10 . 0 kΩ

using a Wheatstone bridge that has

\frac{R_{2}}{R_{1}} = 2 . 000

. Over what range should

R_{3}

be adjustable?

Answer

Range = 5 . 00 Ω to 5.00 k Ω

21-63 §21.6 source ↗

The timing device in an automobile’s intermittent wiper system is based on an

RC

time constant and utilizes a

0 . 500-μF

capacitor and a variable resistor. Over what range must

R

be made to vary to achieve time constants from 2.00 to 15.0 s?

Answer

range 4 . 00 to 30 . 0 M Ω

21-65 §21.6 source ↗

The duration of a photographic flash is related to an

RC

time constant, which is

0 . 100 μs

for a certain camera. (a) If the resistance of the flash lamp is

0.0400 Ω

during discharge, what is the size of the capacitor supplying its energy? (b) What is the time constant for charging the capacitor, if the charging resistance is

800 kΩ

?

Answer

(a) $2 . 50 μF$

(b) 2.00 s

21-67 §21.6 source ↗

After two time constants, what percentage of the final voltage, emf, is on an initially uncharged capacitor

C

, charged through a resistance

R

?

Answer

86.5%

21-69 §21.6 source ↗

A heart defibrillator being used on a patient has an

RC

time constant of 10.0 ms due to the resistance of the patient and the capacitance of the defibrillator. (a) If the defibrillator has an

8 . 00-μF

capacitance, what is the resistance of the path through the patient? (You may neglect the capacitance of the patient and the resistance of the defibrillator.) (b) If the initial voltage is 12.0 kV, how long does it take to decline to

6.00 \times 10^{2} V

?

Answer

(a) $1 . 25 k Ω$

(b) 30.0 ms

21-71 §21.6 source ↗

Figure 21.53 shows how a bleeder resistor is used to discharge a capacitor after an electronic device is shut off, allowing a person to work on the electronics with less risk of shock. (a) What is the time constant? (b) How long will it take to reduce the voltage on the capacitor to 0.250% (5% of 5%) of its full value once discharge begins? (c) If the capacitor is charged to a voltage

V_{0}

through a

100-Ω

resistance, calculate the time it takes to rise to

0.865 V_{0}

(This is about two time constants.)

A parallel circuit with a switch, an embedded electronic circuit, a capacitor, and a resistor is shown. The embedded circuit, capacitor, and resistor are connected in parallel with each other: the electronic circuit on the left, the capacitor in the middle, and the resistor on the right. The capacitor has a capacitance of eighty micro farads. The resistor has a resistance of two hundred fifty kilohms. The switch is on the top, between the electronic circuit and the capacitor leg. — Figure 21.53 in the book ↗

Answer

(a) 20.0 s

(b) 120 s

(c) 16.0 ms

21-73 §21.6 source ↗

Using the exact exponential treatment, find how much time is required to charge an initially uncharged 100-pF capacitor through a

75.0 -M Ω

resistor to 90.0% of its final voltage.

Answer

1 . 73 \times 10^{- 2} s

21-74 §21.6 source ↗

Integrated Concepts

If you wish to take a picture of a bullet traveling at 500 m/s, then a very brief flash of light produced by an $RC$ discharge through a flash tube can limit blurring. Assuming 1.00 mm of motion during one $RC$ constant is acceptable, and given that the flash is driven by a $600-μF$ capacitor, what is the resistance in the flash tube?

Answer

3.33 \times 10^{- 3} Ω

21-76 §21.6 source ↗

Integrated Concepts

A $160-μF$ capacitor charged to 450 V is discharged through a $31 . 2-k Ω$ resistor. (a) Find the time constant. (b) Calculate the temperature increase of the resistor, given that its mass is 2.50 g and its specific heat is $1.67 \frac{kJ}{kg \cdot ºC}$ , noting that most of the thermal energy is retained in the short time of the discharge. (c) Calculate the new resistance, assuming it is pure carbon. (d) Does this change in resistance seem significant?

Answer

(a) 4.99 s

(b) $3 . 87ºC$

(c) $31 . 1 k Ω$

(d) No

21-80 §21.6 source ↗

Critical Thinking A circuit has a voltage source producing

1.00 \times 10^{2} V

and a

2.50 \times 10^{3} Ω

resistor. (a) If the circuit contains nothing else, how much power is dissipated by the resistor? (b) A second

2.50 \times 10^{3} Ω

resistor is added in parallel to the first resistor. Now how much power is dissipated by the resistors? (c) Now place the two resistors in series. How much power is dissipated by the resistors? (d) Why would a parallel arrangement of resistors dissipate more power than a series arrangement of resistors of the same value? (e) Which delivers more energy in a given time period, the parallel combination of resistors, the series combination of resistors, or is the energy delivered in a given time period the same?

Answer

(a) $P =$ $\frac{V^{2}}{R} =$ $\frac{{(1.00 \times 10^{2})}^{2}}{2.50 \times 10^{3}} w =$ $4.00 W$

(b) $\begin{array}{l} \frac{1}{R_{e q}} = \frac{2}{R_{1}} \\ R_{e q} = \frac{R_{1}}{2} = 1.25 \times 10^{3} Ω \\ P = \frac{V^{2}}{R} = \frac{{(1.00 \times 10^{2})}^{2}}{1.25 \times 10^{3}} W = 8 W \end{array}$

(c) $\begin{array}{l} R_{e q} = R_{1} + R_{2} = 2 R_{1} \\ P = \frac{V^{2}}{R} = \frac{{(1.00 \times 10^{2})}^{2}}{5.00 \times 10^{3}} W = 2 W \end{array}$

(d) In the parallel case current passes through each branch, so more current is passed through the resistor system. In the series, the higher resistance of the circuit allows less current to flow. $P = I V$ is proportional to the current when the voltage does not change.

(e) The parallel combination delivers more energy in a given time period.

AP Test Prep 21.1 §21.1 source ↗

The figure above shows a circuit containing two batteries and three identical resistors with resistance R. Which of the following changes to the circuit will result in an increase in the current at point P? Select two answers.

a.
Reversing the connections to the 14 V battery.
b.
Removing the 2 V battery and connecting the wires to close the left loop.
c.
Rearranging the resistors so all three are in series.
d.
Removing the branch containing resistor Z.

A circuit is shown as a rectangle with an extra line connecting the top and bottom having resistor Y. The top of this circuit has a 2-volt battery to the left of the Y resistor line, the left side of this circuit has a point P marked and then a 14-volt battery, the bottom has a resistor X to the left of the Y resistor line, and the right side has a resistor Z. — Figure 21.54 in the book ↗

Answer

A. Reversing the connections to the 14 V battery. · B. Removing the 2 V battery and connecting the wires to close the left loop.

AP Test Prep 21.3 §21.1 source ↗

If the circuit in the previous question is modified by removing some of the 1.6 kΩ resistors, the total current in the circuit is 24 mA. How many resistors were removed?

a.
1
b.
2
c.
3
d.
4

Answer

B. 2

AP Test Prep 21.5 §21.1 source ↗

In a circuit, a parallel combination of two 20-Ω and one 10-Ω resistors is connected in series with a 4-Ω resistor. The source voltage is 36 V.

(a) Find the resistor(s) with the maximum current.

(b) Find the resistor(s) with the maximum voltage drop.

(c) Find the power dissipated in each resistor and hence the total power dissipated in all the resistors. Also find the power output of the source. Are they equal or not? Justify your answer.

(d) Will the answers for questions (a) and (b) differ if a 3 Ω resistor is added in series to the 4 Ω resistor? If yes, repeat the question(s) for the new resistor combination.

(e) If the values of all the resistors and the source voltage are doubled, what will be the effect on the current?

Answer

(a) 4-Ω resistor; (b) combination of 20-Ω, 20-Ω, and 10-Ω resistors; (c) 20 W in each 20-Ω resistor, 40 W in 10-Ω resistor, 64 W in 4-Ω resistor, total 144W total in resistors, output power is 144 W, yes they are equal (law of conservation of energy); (d) 4 Ω and 3 Ω for part (a) and no change for part (b); (e) no effect, it will remain the same.

AP Test Prep 21.7 §21.2 source ↗

Calculate the internal resistance of a voltage source if the terminal voltage of the source increases by 1 V when the current supplied decreases by 4 A? Suppose this source is connected in series (in the same direction) to another source with a different voltage but same internal resistance. What will be the total internal resistance? How will the total internal resistance change if the sources are connected in the opposite direction?

Answer

0.25 Ω, 0.50 Ω, no change

AP Test Prep 21.9 §21.3 source ↗

Circuit that across the top from left to right goes point b, battery with voltage E1, point c, resistor with resistance r1, and point d; across the middle goes point a, battery with voltage E2, point k, resistor with resistance r2, point l, resistor with resistance R2, and point e; across the bottom goes point j, battery with voltage E3, point i, resistor with resistance r3, and point h; along the left side from top to bottom goes a resistor with resistance R1, point a, and a resistor with resistance R3; and along the right side goes a resistor with resistance R5, point e, a resistor with resistance r4, point f, a battery with voltage E4, and point g. Additionally, there are three arrows showing the direction of the current: one between point a and the resistor with resistance R1 pointing up; another between point a and the battery with voltage E2 pointing right; and another between point j and the resistor with resistance R3 pointing up. — Figure 21.62 in the book ↗

For this question, consider the circuit shown in the following figure.

a.

Assuming that none of the three currents (I₁, I₂, and I₃) are equal to zero, which of the following statements is false?

(a) I₃ = I₁ + I₂ at point a.

(b) I₂ = I₃ - I₁ at point e.

(c) The current through R₃ is equal to the current through R_5.

(d) The current through R₁ is equal to the current through R_5.

b.

Which of the following statements is true?

(a) E₁ + E₂ + I₁R₁ - I₂R₂ + I₁r₁ - I₂r₂ + I₁R₅ = 0

(b) - E₁ + E₂ + I₁R₁ - I₂R₂ + I₁r₁ - I₂r₂ - I₁R₅ = 0

(c) E₁ - E₂ - I₁R₁ + I₂R₂ - I₁r₁ + I₂r₂ - I₁R₅ = 0

(d) E₁ + E₂ - I₁R₁ + I₂R₂ - I₁r₁ + I₂r₂ + I₁R₅ = 0

c.

If I₁ = 5 A and I₃ = -2 A, which of the following statements is false?

(a) The current through R₁ will flow from a to b and will be equal to 5 A.

(b) The current through R₃ will flow from a to j and will be equal to 2 A.

(c) The current through R₅ will flow from d to e and will be equal to 5 A.

(d) None of the above.

d.

If I₁ = 5 A and I₃ = -2 A, I₂ will be equal to

(a) 3 A

(b) -3 A

(c) 7 A

(d) -7 A

Answer

a. (c)

b. (c)

c. (d)

d. (d)

AP Test Prep 21.11 §21.3 source ↗

Circuit with a battery with voltage E1 and resistor with resistance r1 across the top from left to right; point A, a resistor with resistance R1, and point B across the middle; and a battery marked E2 and a resistor with resistance r2 across the bottom. Additionally, on the left, from top to bottom there is a resistor with resistance R2 and point A; on the right, from top to bottom there is point B and a resistor with resistance R3. — Figure 21.65 in the book ↗

In this circuit, assume the currents through R₁, R₂ and R₃ are I₁, I₂ and I₃ respectively and all are flowing in the clockwise direction.

(a) Find the equation obtained by applying Kirchhoff’s junction rule at point A.

(b) Find the equations obtained by applying Kirchhoff’s loop rule in the upper and lower loops.

(c) Assume R₁ = R₂ = 6 Ω, R₃ = 12 Ω, r₁ = r₂ = 0 Ω, E₁ = 6 V and E₂ = 4 V. Calculate I₁, I₂ and I₃.

(d) For the situation in which E₂ is replaced by a closed switch, repeat parts (a) and (b). Using the values for R₁, R₂, R₃, r₁ and E₁ from part (c) calculate the currents through the three resistors.

(e) For the circuit in part (d) calculate the output power of the voltage source and across all the resistors. Examine if energy is conserved in the circuit.

(f) A student implemented the circuit of part (d) in the lab and measured the current though one of the resistors as 0.19 A. According to the results calculated in part (d) identify the resistor(s). Justify any difference in measured and calculated value.

Answer

(a) I₁ + I₃ = I₂

(b) E₁ - I₁R₁ - I₂R₂ - I₁r₁ = 0; - E₂ + I₁R₁ - I₃R₃ - I₃r₂ = 0

(c) I₁ = 8/15 A, I₂ = 7/15 A and I₃ = -1/15 A

(d) I₁ = 2/5 A, I₂ = 3/5 A and I₃ = 1/5 A

(e) P_E1 = 18/5 W and P_R1 = 24/25 W, P_R2 = 54/25 W, P_R3 = 12/25 W. Yes, P_E1 = P_R1+ P_R2 + P_R3

(f) R_3, losses in the circuit

AP Test Prep 21.13 §21.6 source ↗

An uncharged capacitor C is connected in series (with a switch) to a resistor R₁ and a voltage source E. Assume E = 24 V, R₁ = 1.2 kΩ and C = 1 mF.

(a) What will be the current through the circuit as the switch is closed? Draw a circuit diagram and show the direction of current after the switch is closed. How long will it take for the capacitor to be 99% charged?

(b) After full charging, this capacitor is connected in series to another resistor, R₂ = 1 kΩ. What will be the current in the circuit as soon as it’s connected? Draw a circuit diagram and show the direction of current. How long will it take for the capacitor voltage to reach 3.24 V?

Answer

(a) 20 mA, Figure 21.37, 5.5 s; (b) 24 mA, Figure 21.29, 2 s

Part a shows a circuit with a cell of e m f script E connected in series with a resistor R, a capacitor C, and a switch to close the circuit. The current is shown flowing in a clockwise direction. The capacitor plates are shown to have a charge positive q and negative q respectively. Part b shows a graph of the variation of voltage of the capacitor with time. The voltage is plotted along the vertical axis and the time is along the horizontal axis. The graph shows a smooth upward rising curve which approaches a maximum and flattens out at maximum voltage equal to e m f script E over time. — Figure 21.37 in the book ↗

This figure has two parts, each of which shows two rough surfaces in close proximity to each other. In the first part, the normal force is small, so that the area of contact between the two surfaces is much smaller than their total area. In the second part, the normal force is large, so that the area of contact between the two surfaces has increased. As a result, the friction between the two surfaces in the second part is also greater than the friction in the first part. — Figure 5.5 in the book ↗

The drawing shows a voltmeter, which is a circuit with a large resistance in series with a galvanometer, along with its internal resistance. — Figure 21.29 in the book ↗