Chapter 21: Circuits and DC InstrumentsContent check — College Physics, 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 $\text{275-Ω}$ resistors connected in series? (b) In parallel?

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 $\text{36}\text{.}\mathrm{0-\Omega }$, a $\text{50}\text{.}\mathrm{0-\Omega }$, and a $\text{700-Ω}$ resistor together?

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.)

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.

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 $0\text{.}\text{400}\Omega$, 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?

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 ($R_1\text{>>}{R}_2$): (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 $R_2$.

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

Unreasonable Results

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

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?

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 $2\text{.}\text{00}\Omega$?

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?

The hot resistance of a flashlight bulb is $2\text{.}\text{30}\Omega$, and it is run by a 1.58-V alkaline cell having a $0\text{.}\text{100-Ω}$ internal resistance. (a) What current flows? (b) Calculate the power supplied to the bulb using $I^2{R}_{\text{bulb}}$. (c) Is this power the same as calculated using $\frac{V^2}{R_{\text{bulb}}}$?

An automobile starter motor has an equivalent resistance of $0\text{.}\text{0500}\Omega$ and is supplied by a 12.0-V battery with a $0\text{.}\text{0100-Ω}$ 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 $0\text{.}\text{0900}\Omega$ to the circuit. (Significant problems are caused by even small amounts of unwanted resistance in low-voltage, high-current applications.)

(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?

Unreasonable Results

A 1.58-V alkaline cell with a $0\text{.}\text{200-Ω}$ 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?

Apply the loop rule to loop abcdefgha in Figure 21.25.

Apply the junction rule at point a in Figure 21.50.

Apply the loop rule to loop akledcba in Figure 21.50.

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.

Unreasonable Results

Consider the circuit in Figure 21.51, and suppose that the emfs are unknown and the currents are given to be $I_1=5\text{.}\text{00 A}$, $I_2=3\text{.0 A}$, and $I_3=\mathrm{–2}\text{.}\text{00 A}$. (a) Could you find the emfs? (b) What is wrong with the assumptions?

What is the sensitivity of the galvanometer (that is, what current gives a full-scale deflection) inside a voltmeter that has a $1\text{.}\text{00}\text{-M}\Omega$ resistance on its 30.0-V scale?

Find the resistance that must be placed in series with a $\text{25}\text{.}\mathrm{0-\Omega }$ galvanometer having a $\text{50.0}\mathrm{-\mu A}$ 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.

Find the resistance that must be placed in parallel with a $\text{25}\text{.}\mathrm{0-\Omega }$ galvanometer having a $\text{50}\text{.}\mathrm{0-\mu A}$ 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.

Find the resistance that must be placed in series with a $\text{10}\text{.}\mathrm{0-\Omega }$ galvanometer having a $\text{100-μA}$ 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.

Suppose you measure the terminal voltage of a 1.585-V alkaline cell having an internal resistance of $0\text{.}\text{100}\Omega$ by placing a $1\text{.}\text{00}\text{-k}\Omega$ 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.

A certain ammeter has a resistance of $5\text{.}\text{00}\times {\text{10}}^{-5}\Omega$ on its 3.00-A scale and contains a $\text{10}\text{.}\mathrm{0-\Omega }$ galvanometer. What is the sensitivity of the galvanometer?

A $0\text{.}\text{0200-Ω}$ ammeter is placed in series with a $\text{10}\text{.}\text{00-Ω}$ 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 $\text{10}\text{.}\text{00-Ω}$ 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 $\text{10}\text{.}\text{00-Ω}$ resistor alone, what is the percent increase in voltage? (e) Are the changes found in parts (c) and (d) significant? Discuss.

Unreasonable Results

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

What is the ${\text{emf}}_{\text{x}}$ of a cell being measured in a potentiometer, if the standard cell’s emf is 12.0 V and the potentiometer balances for $R_{\text{x}}=5\text{.}\text{000}\Omega$ and $R_{\text{s}}=2\text{.}\text{500}\Omega$?

When an unknown resistance $R_{\text{x}}$ is placed in a Wheatstone bridge, it is possible to balance the bridge by adjusting $R_3$ to be $\text{2500}\Omega$. What is $R_{\text{x}}$ if $\frac{R_2}{R_1}=0\text{.}\text{625}$?

(a) What is the unknown ${\text{emf}}_{\text{x}}$ in a potentiometer that balances when $R_{\text{x}}$ is $\text{10}\text{.}0\Omega$, and balances when $R_{\text{s}}$ is $\text{15}\text{.}0\Omega$ for a standard 3.000-V emf? (b) The same ${\text{emf}}_{\text{x}}$ is placed in the same potentiometer, which now balances when $R_{\text{s}}$ is $\text{15}\text{.}0\Omega$ for a standard emf of 3.100 V. At what resistance $R_{\text{x}}$ will the potentiometer balance?

Suppose you want to measure resistances in the range from $\text{10}\text{.}0\Omega$ to $\text{10}\text{.}\mathrm{0\; k\Omega }$ using a Wheatstone bridge that has $\frac{R_2}{R_1}=2\text{.}\text{000}$. Over what range should $R_3$ be adjustable?

The timing device in an automobile’s intermittent wiper system is based on an $\text{RC}$ time constant and utilizes a $0\text{.}\text{500-μF}$ 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?

The duration of a photographic flash is related to an $\text{RC}$ time constant, which is $0\text{.}\text{100 μs}$ for a certain camera. (a) If the resistance of the flash lamp is $0\text{.}\text{0400}\Omega$ 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 $\text{800}\text{kΩ}$?

After two time constants, what percentage of the final voltage, emf, is on an initially uncharged capacitor $C$, charged through a resistance $R$?

A heart defibrillator being used on a patient has an $\text{RC}$ 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 $8\text{.}\text{00-μF}$ 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 $6.00\times {10}^2\text{V}$?

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 $\text{100-Ω}$ resistance, calculate the time it takes to rise to $0\text{.}\text{865}{V}_0$ (This is about two time constants.)

Using the exact exponential treatment, find how much time is required to charge an initially uncharged 100-pF capacitor through a $\text{75}\text{.}0\text{-M}\Omega$ resistor to 90.0% of its final voltage.

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 $\text{RC}$ discharge through a flash tube can limit blurring. Assuming 1.00 mm of motion during one $\text{RC}$ constant is acceptable, and given that the flash is driven by a $\text{600-μF}$ capacitor, what is the resistance in the flash tube?

Integrated Concepts

A $\text{160-μF}$ capacitor charged to 450 V is discharged through a $31\text{.}\text{2-k}\Omega$ 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 $1\text{.}\text{67}\frac{\text{kJ}}{\text{kg}\cdot \text{ºC}}$, 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?

Critical Thinking A circuit has a voltage source producing $1.00\times {10}^2\text{V}$ and a $2.50\times {10}^3\text{Ω}$ resistor. (a) If the circuit contains nothing else, how much power is dissipated by the resistor? (b) A second $2.50\times {10}^3\text{Ω}$ 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?