### Problems

## 10.1 Electromotive Force

A car battery with a 12-V emf and an internal resistance of $0.050\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ is being charged with a current of 60 A. Note that in this process, the battery is being charged. (a) What is the potential difference across its terminals? (b) At what rate is thermal energy being dissipated in the battery? (c) At what rate is electric energy being converted into chemical energy?

The label on a battery-powered radio recommends the use of a rechargeable nickel-cadmium cell (nicads), although it has a 1.25-V emf, whereas an alkaline cell has a 1.58-V emf. The radio has a $3.20\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ resistance. (a) Draw a circuit diagram of the radio and its battery. Now, calculate the power delivered to the radio (b) when using a nicad cells, each having an internal resistance of $0.0400\phantom{\rule{0.2em}{0ex}}\text{\Omega}$, and (c) when using an alkaline cell, having an internal resistance of $0.200\phantom{\rule{0.2em}{0ex}}\text{\Omega}$. (d) Does this difference seem significant, considering that the radio’s effective resistance is lowered when its volume is turned up?

An automobile starter motor has an equivalent resistance of $0.0500\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ and is supplied by a 12.0-V battery with a $0.0100\text{-}\text{\Omega}$ 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 $0.0900\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ 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 potential 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?

A person with body resistance between their hands of $10.0\phantom{\rule{0.2em}{0ex}}\text{k}\text{\Omega}$ accidentally grasps the terminals of a 20.0-kV power supply. (Do NOT do this!) (a) Draw a circuit diagram to represent the situation. (b) If the internal resistance of the power supply is $2000\phantom{\rule{0.2em}{0ex}}\text{\Omega}$, what is the current through their body? (c) What is the power dissipated in their body? (d) If the power supply is to be made safe by increasing its internal resistance, what should the internal resistance be for the maximum current in this situation to be 1.00 mA or less? (e) Will this modification compromise the effectiveness of the power supply for driving low-resistance devices? Explain your reasoning.

A 12.0-V emf automobile battery has a terminal voltage of 16.0 V when being charged by a current of 10.0 A. (a) What is the battery’s internal resistance? (b) What power is dissipated inside the battery? (c) At what rate (in $\text{\xb0}\text{C}\text{/}\text{min}$) will its temperature increase if its mass is 20.0 kg and it has a specific heat of $0.300\phantom{\rule{0.2em}{0ex}}\text{kcal/kg}\phantom{\rule{0.2em}{0ex}}\xb7\text{\xb0}\text{C}$, assuming no heat escapes?

## 10.2 Resistors in Series and Parallel

(a) What is the resistance of a $1.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}\text{-}\text{\Omega}$, a $2.50\text{-k}\text{\Omega}$, and a $4.00\text{-k}\text{\Omega}$ resistor connected in series? (b) In parallel?

What are the largest and smallest resistances you can obtain by connecting a $36.0\text{-}\text{\Omega}$, a $50.0\text{-}\text{\Omega}$, and a $700\text{-}\text{\Omega}$ resistor together?

An 1800-W toaster, a 1400-W speaker, and a 75-W lamp are plugged into the same outlet in a 15-A fuse and 120-V circuit. (The three devices are in parallel when plugged into the same socket.) (a) What current is drawn by each device? (b) Will this combination blow the 15-A fuse?

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.)

(a) Given a 48.0-V battery and $24.0\text{-}\text{\Omega}$ and $96.0\text{-}\text{\Omega}$ resistors, find the current and power for each when connected in series. (b) Repeat when the resistances are in parallel.

Referring to the example combining series and parallel circuits and Figure 10.16, 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 Series and Parallel Resistors.

Referring to Figure 10.16, (a) Calculate ${P}_{3}$ and note how it compares with ${P}_{3}$ found in the first two example problems in this module. (b) Find the total power supplied by the source and compare it with the sum of the powers dissipated by the resistors.

Refer to Figure 10.17 and the discussion of lights dimming when a heavy appliance comes on. (a) Given the voltage source is 120 V, the wire resistance is $0.800\phantom{\rule{0.2em}{0ex}}\text{\Omega ,}$ 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?

Show that if two resistors ${R}_{1}$ and ${R}_{2}$ are combined and one is much greater than the other $({R}_{1}\gg {R}_{2})$, (a) their series resistance is very nearly equal to the greater resistance${R}_{1}$ and (b) their parallel resistance is very nearly equal to the smaller resistance ${R}_{2}$.

Consider the circuit shown below. The terminal voltage of the battery is $V=18.00\phantom{\rule{0.2em}{0ex}}\text{V}.$ (a) Find the equivalent resistance of the circuit. (b) Find the current through each resistor. (c) Find the potential drop across each resistor. (d) Find the power dissipated by each resistor. (e) Find the power supplied by the battery.

## 10.3 Kirchhoff's Rules

Consider the circuit shown below. (a) Find the voltage across each resistor. (b)What is the power supplied to the circuit and the power dissipated or consumed by the circuit?

Consider the circuits shown below. (a) What is the current through each resistor in part (a)? (b) What is the current through each resistor in part (b)? (c) What is the power dissipated or consumed by each circuit? (d) What is the power supplied to each circuit?

Consider the circuit shown below. Find ${V}_{1},{I}_{2},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{I}_{3}.$

Consider the circuit shown below. Find ${V}_{1},{V}_{2},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{R}_{4}.$

Consider the circuit shown below. Find ${I}_{1},{I}_{2},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{I}_{3}.$

Consider the circuit shown below. (a) Find ${I}_{1},{I}_{2},{I}_{3},{I}_{4},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{I}_{5}.$ (b) Find the power supplied by the voltage sources. (c) Find the power dissipated by the resistors.

Consider the circuit shown below. Write the three loop equations for the loops shown.

Consider the circuit shown in the preceding problem. Write equations for the power supplied by the voltage sources and the power dissipated by the resistors in terms of *R* and *V*.

A child’s electronic toy is supplied by three 1.58-V alkaline cells having internal resistances of $0.0200\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ in series with a 1.53-V carbon-zinc dry cell having a $0.100\text{-}\text{\Omega}$ internal resistance. The load resistance is $10.0\phantom{\rule{0.2em}{0ex}}\text{\Omega}$. (a) Draw a circuit diagram of the toy and its batteries. (b) What current flows? (c) How much power is supplied to the load? (d) What is the internal resistance of the dry cell if it goes bad, resulting in only 0.500 W being supplied to the load?

Apply the junction rule to Junction *b* shown below. Is any new information gained by applying the junction rule at *e*?

## 10.4 Electrical Measuring Instruments

Suppose you measure the terminal voltage of a 1.585-V alkaline cell having an internal resistance of $0.100\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ by placing a $1.00\text{-k}\text{\Omega}$ voltmeter across its terminals (see below). (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.

## 10.5 RC Circuits

The timing device in an automobile’s intermittent wiper system is based on an *RC* time constant and utilizes a $0.500\text{-}\mu \text{F}$ 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?

A heart pacemaker fires 72 times a minute, each time a 25.0-nF capacitor is charged (by a battery in series with a resistor) to 0.632 of its full voltage. What is the value of the resistance?

The duration of a photographic flash is related to an *RC* time constant, which is $0.100\mu \text{s}$ for a certain camera. (a) If the resistance of the flash lamp is $0.0400\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ 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 $800\phantom{\rule{0.2em}{0ex}}\text{k}\text{\Omega}$?

A 2.00- and a $7.50\text{-}\mu \text{F}$ capacitor can be connected in series or parallel, as can a 25.0- and a $100\text{-k}\text{\Omega}$ resistor. Calculate the four *RC* time constants possible from connecting the resulting capacitance and resistance in series.

A $500\text{-}\text{\Omega}$ resistor, an uncharged $1.50\text{-}\mu \text{F}$ capacitor, and a 6.16-V emf are connected in series. (a) What is the initial current? (b) What is the *RC* time constant? (c) What is the current after one time constant? (d) What is the voltage on the capacitor after one time constant?

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 a capacitance of $8.00\mu \text{F},$ 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 $6.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}\phantom{\rule{0.2em}{0ex}}\text{V}$?

An ECG monitor must have an *RC* time constant less than $1.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}\mu \text{s}$ to be able to measure variations in voltage over small time intervals. (a) If the resistance of the circuit (due mostly to that of the patient’s chest) is $1.00\phantom{\rule{0.2em}{0ex}}\text{k}\text{\Omega}$, what is the maximum capacitance of the circuit? (b) Would it be difficult in practice to limit the capacitance to less than the value found in (a)?

Using the exact exponential treatment, determine how much time is required to charge an initially uncharged 100-pF capacitor through a $75.0\text{-M}\text{\Omega}$ resistor to $90.0\%$ of its final voltage.

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 $600\text{-}\mu \text{F}$ capacitor, what is the resistance in the flash tube?

## 10.6 Household Wiring and Electrical Safety

(a) How much power is dissipated in a short circuit of 240-V ac through a resistance of $0.250\phantom{\rule{0.2em}{0ex}}\text{\Omega}$? (b) What current flows?

What voltage is involved in a 1.44-kW short circuit through a $0.100\text{-}\text{\Omega}$ resistance?

Find the current through a person and identify the likely effect on her if she touches a 120-V ac source: (a) if she is standing on a rubber mat and offers a total resistance of $4000\phantom{\rule{0.2em}{0ex}}\text{k}\text{\Omega}$; (b) if she is standing barefoot on wet grass and has a resistance of only $300\phantom{\rule{0.2em}{0ex}}\text{k}\text{\Omega}$.

While taking a bath, a person touches the metal case of a radio. The path through the person to the drainpipe and ground has a resistance of $4000\phantom{\rule{0.2em}{0ex}}\text{\Omega}$. What is the smallest voltage on the case of the radio that could cause ventricular fibrillation?

A man foolishly tries to fish a burning piece of bread from a toaster with a metal butter knife and comes into contact with 120-V ac. He does not even feel it since, luckily, he is wearing rubber-soled shoes. What is the minimum resistance of the path the current follows through the person?

(a) During surgery, a current as small as $20.0\phantom{\rule{0.2em}{0ex}}\mu \text{A}$ applied directly to the heart may cause ventricular fibrillation. If the resistance of the exposed heart is $300\phantom{\rule{0.2em}{0ex}}\text{\Omega},$ what is the smallest voltage that poses this danger? (b) Does your answer imply that special electrical safety precautions are needed?

(a) What is the resistance of a 220-V ac short circuit that generates a peak power of 96.8 kW? (b) What would the average power be if the voltage were 120 V ac?

A heart defibrillator passes 10.0 A through a patient’s torso for 5.00 ms in an attempt to restore normal beating. (a) How much charge passed? (b) What voltage was applied if 500 J of energy was dissipated? (c) What was the path’s resistance? (d) Find the temperature increase caused in the 8.00 kg of affected tissue.

A short circuit in a 120-V appliance cord has a $0.500\text{-}\text{\Omega}$ resistance. Calculate the temperature rise of the 2.00 g of surrounding materials, assuming their specific heat capacity is $0.200\phantom{\rule{0.2em}{0ex}}\text{cal/g}\xb7\text{\xb0}\text{C}$ and that it takes 0.0500 s for a circuit breaker to interrupt the current. Is this likely to be damaging?