Chapter 19

Typical household currents are on the order of a few amperes. If a 1.50 A current flows through the leads of an electrical appliance, (a) how many electrons per second pass through it, (b) how many coulombs pass through it in 5.0 min, and (c) how long does it take for 7.50 \(\mathrm{C}\) of charge to pass through?

Lightning strikes. During lightning strikes from a cloud to the ground, currents as high as \(25,000\) A can occur and last for about 40\(\mu\) s. How much charge is transferred from the cloud to the earth during such a strike?

Transmission of nerve impulses. Nerve cells transmit electric signals through their long tubular axons. These signals propagate due to a sudden rush of \(\mathrm{Na}^{+}\) ions, each with charge \(+e,\) into the axon. Measurements have revealed that typically about \(5.6 \times 10^{11} \mathrm{Na}^{+}\) ions enter each meter of the axon during a time of 10 \(\mathrm{ms}\) . What is the current during this inflow of charge in a meter of axon?

In an ionic solution, a current consists of \(\mathrm{Ca}^{2+}\) ions (of charge \(+2 e )\) and \(\mathrm{Cl}^{-}\) ions (of charge \(-e )\) traveling in opposite directions. If \(5.11 \times 10^{18} \mathrm{Cl}^{-}\) ions go from \(A\) to \(B\) every 0.50 min, while \(3.24 \times 10^{18} \mathrm{Ca}^{2+}\) ions move from \(B\) to \(A\), what is the current (in mA) through this solution, and in which direction \((\) from \(A\) to \(B\) or from \(B\) to \(A)\) is it going?

Copper has \(8.5 \times 10^{28}\) electrons per cubic meter. (a) How many electrons are there in a 25.0 \(\mathrm{cm}\) length of 12 -gauge copper wire (diameter 2.05 \(\mathrm{mm} ) ?\) (b) If a current of 1.55 \(\mathrm{A}\) is flowing in the wire, what is the average drift speed of the electrons along the wire? (There are \(6.24 \times 10^{18}\) electrons in 1 coulomb of charge.)

A wire 6.50 \(\mathrm{m}\) long with diameter of 2.05 \(\mathrm{mm}\) has a resistance of 0.0290\(\Omega .\) What material is the wire most likely made of?

A tightly coiled spring having 75 coils, each 3.50 \(\mathrm{cm}\) in diameter, is made of insulated metal wire 3.25 \(\mathrm{mm}\) in diameter. An ohmmeter connected across opposite ends of the spring reads 1.74\(\Omega .\) What is the resistivity of the metal?

If you triple the length of a cable and at the same time double its diameter, what will be its resistance if its original resistance was \(R\) ?

A ductile metal wire has resistance \(R .\) What will be the resistance of this wire in terms of \(R\) if it is stretched to three times its original length, assuming that the density and resistivity of the material do not change when the wire is stretched. (Hint: The amount of metal does not change, so stretching out the wire will affect its cross-sectional area.)

What is the resistance of a NichromeTM wire at \(0.0^{\circ} \mathrm{C}\) if its resistance is 100.00\(\Omega\) at \(11.5^{\circ} \mathrm{C}\) ? The temperature coefficient of resistivity for Nichrome is 0.00040\(\left(\mathrm{C}^{\circ}\right)^{-1}\)

A 1.50 -m cylindrical rod of diameter 0.500 \(\mathrm{cm}\) is connected to a power supply that maintains a constant potential difference of 15.0 \(\mathrm{V}\) across its ends, while an ammeter measures the current through it. You observe that at room temperature \(\left(20.0^{\circ} \mathrm{C}\right)\) the ammeter reads 18.5 \(\mathrm{A}\) , while at \(92.0^{\circ} \mathrm{C}\) it reads 17.2 \(\mathrm{A} .\) You can ignore any thermal expansion of the rod. Find (a) the resistivity and (b) the temperature coefficient of resistivity at \(20^{\circ} \mathrm{C}\) for the material of the rod.

A carbon resistor having a temperature coefficient of resistivity of \(-0.00050\left(\mathrm{C}^{\circ}\right)^{-1},\) is to be used as a thermometer. On a winter day when the temperature is \(4.0^{\circ} \mathrm{C},\) the resistance of the carbon resistor is 217.3\(\Omega .\) What is the temperature on a spring day when the resistance is 215.8\(\Omega ?\) (Take the reference temperature \(T_{0}\) to be \(4.0^{\circ} \mathrm{C.}\) .

The following measurements of current and potential difference were made on a resistor constructed of Nichrome \(^{\mathrm{TM}}\) wire, where \(V_{a b}\) is the potential difference across the wire and \(I\) is the current through it: $$\begin{array}{ccccc}{I(\mathrm{A})} & {0.50} & {1.00} & {2.00} & {4.00} \\\ {V_{a b}(\mathrm{V})} & {1.94} & {3.88} & {7.76} & {15.52}\end{array}$$ (a) Graph \(V_{a b}\) as a function of \(I .\) (b) Does Ohm's law apply to Nichromet"y? How can you tell? (c) What is the resistance of the resistor in ohms?

When you connect an unknown resistor across the terminals of a 1.50 \(\mathrm{V}\) AAA battery having negligible internal resistance, you measure a current of 18.0 \(\mathrm{mA}\) flowing through it. (a) What is the resistance of this resistor? (b) If you now place the resistor across the terminals of 12.6 \(\mathrm{V}\) car battery having no inter- nal resistance, how much current will flow? (c) You now put the resistor across the terminals of an unknown battery of negligible internal resistance and measure a current of 0.453 \(\mathrm{A}\) flowing through it. What is the potential difference across the terminals of the battery?

Current in the body. The resistance of the body varies from approximately 500 \(\mathrm{k} \Omega\) (when it is very dry) to about 1 \(\mathrm{k} \Omega\) (when it is wet). The maximum safe current is about 5.0 \(\mathrm{mA} .\) At 10 \(\mathrm{mA}\) or above, muscle contractions can occur that may be fatal. What is the largest potential difference that a person can safely touch if his body is wet? Is this result within the range of common household voltages?

When a solid cylindrical rod is connected across a fixed potential difference, a current \(I\) flows through the rod. What would be the current (in terms of \(I\) ) if (a) the length were doubled, (b) the diameter were doubled, (c) both the length and the diameter were doubled?

A 6.00 \(\mathrm{V}\) lantern battery is connected to a 10.5\(\Omega\) lightbulb, and the resulting current in the circuit is 0.350 A. What is the internal resistance of the battery?

. A complete series circuit consists of a 12.0 \(\mathrm{V}\) battery, a 4.70\(\Omega\) resistor, and a switch. The internal resistance of the battery is 0.30\(\Omega .\) The switch is open. What does an ideal voltmeter read when placed (a) across the terminals of the battery, (b) across the resistor, (c) across the switch? (d) Repeat parts (a), (b), and (c) for the case when the switch is closed.

With a 1500 \(\mathrm{M\Omega}\) resistor across its terminals, the terminal voltage of a certain battery is 2.50 \(\mathrm{V}\) . With only a 5.00\(\Omega\) ? resistor across its terminals, the terminal voltage is 1.75 \(\mathrm{V}\) . (a) Find the internal emf and the internal resistance of this battery. (b) What would be the terminal voltage if the 5.00\(\Omega\) resistor were replaced by a 7.00 \(\Omega\) resistor?

An automobile starter motor is connected to a 12.0 \(\mathrm{V}\) battery. When the starter is activated it draws 150 \(\mathrm{A}\) of current, and the battery voltage drops to 7.0 \(\mathrm{V} .\) What is the battery's internal resistance?

A resistor with a 15.0 \(\mathrm{V}\) potential difference across its ends develops thermal energy at a rate of 327 \(\mathrm{W}\) . (a) What is the current in the resistor? (b) What is its resistance?

Power rating of a resistor. The power rating of a resistor is the maximum power it can safely dissipate without being damaged by overheating. (a) If the power rating of a certain 15 \(\mathrm{k} \Omega\) resistor is \(5.0 \mathrm{W},\) what is the maximum current it can carry without damage? What is the greatest allowable potential difference across the terminals of this resistor? (b) If a 9.0 \(\mathrm{k} \Omega\) resistor is to be connected across a 120 \(\mathrm{V}\) potential difference, what power rating is required for that resistor?

An idealized voltmeter is connected across the terminals of a 15.0 \(\mathrm{V}\) battery, and a 75.0\(\Omega\) appliance is also connected across its terminals. If the voltmeter reads \(11.3 \mathrm{V} :\) (a) how much power is being dissipated by the appliance, and (b) what is the internal resistance of the battery?

Treatment of heart failure. A heart defibrillator is used to enable the heart to start beating if it has stopped. This is done by passing a large current of 12 A through the body at 25 \(\mathrm{V}\) for a very short time, usually about 3.0 \(\mathrm{ms}\) (a) What power does the defibrillator deliver to the body, and (b) how much energy is transferred?

Lightbulbs. The wattage rating of a lightbulb is the power it consumes when it is connected across a 120 potential difference. For example, a 60 W lightbulb consumes 60.0 \(\mathrm{W}\) of electrical power only when it is connected across a 120 \(\mathrm{V}\) potential difference. (a) What is the resistance of a 60 \(\mathrm{W}\) lightbulb? (b) Without doing any calculations, would you expect a 100 \(\mathrm{W}\) bulb to have more or less resistance than a 60 \(\mathrm{W}\) bulb? Calculate and find out.

Electrical safety. This procedure is not recommended! You'll see why after you work the problem. You are on an aluminum ladder that is standing on the ground, trying to fix an electrical connection with a metal screwdriver having a metal handle. Your body is wet because you are sweating from the exertion; therefore, it has a resistance of 1.0 \(\mathrm{k} \Omega\) . (a) If you accidentally touch the "hot" wire connected to the 120 \(\mathrm{V}\) line, how much current will pass through your body? Is this amount enough to be dangerous? (The maximum safe current is about 5 \(\mathrm{mA}\). (b) How much electrical power is delivered to your body?

Electric eels. Electric eels generate electric pulses along their skin that can be used to stun an enemy when they come into contact with it. Tests have shown that these pulses can be up to 500 \(\mathrm{V}\) and produce currents of 80 \(\mathrm{mA}\) (or even larger). A typical pulse lasts for 10 \(\mathrm{ms}\) . What power and how much energy are delivered to the unfortunate enemy with a single pulse, assuming a steady current?

Electric space heater. A "540 W" electric heater is designed to operate from 120 V lines. (a) What is its resistance, and (b) what current does it draw? (c) At 7.4\(\notin\) per kWh, how much does it cost to operate this heater for an hour? (d) If the line voltage drops to \(110 \mathrm{V},\) what power does the heater take, in watts? (Assume that the resistance is constant, although it actually will change because of the change in temperature.)

The battery for a certain cell phone is rated at 3.70 \(\mathrm{V}\) . According to the manufacturer it can produce \(3.15 \times 10^{4} \mathrm{J}\) of electrical energy, enough for 5.25 \(\mathrm{h}\) of operation, before needing to be recharged. Find the average current that this cell phone draws when turned on.

A \(540-\mathrm{W}\) electric heater is designed to operate from 120 \(\mathrm{V}\) lines. (a) What is its resistance? (b) What current does it draw? (c) If the line voltage drops to \(110 \mathrm{V},\) what power does the heater take? (Assume that the resistance is constant. Actually, it will change because of the change in temperature.) (d) The heater coils are metallic, so that the resistance of the heater decreases with decreasing temperature. If the change of resistance with temperature is taken into account, will the electrical power consumed by the heater be larger or smaller than what you calculated in part (c)? Explain.

Electricity through the body, I. A person with a body resistance of 10 \(\mathrm{k} \Omega\) between his hands accidentally grasps the terminals of a 14 \(\mathrm{kV}\) power supply. (a) If the internal resistance of the power supply is \(2000 \Omega,\) what is the current through the person's body? (b) What is the power dissipated in his body? (c) 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 the situation just described to be 1.00 \(\mathrm{mA}\) or less?

Electricity through the body, II. The average bulk resistivity of the human body (apart from surface resistance of the skin) is about 5.0\(\Omega \cdot \mathrm{m} .\) The conducting path between the hands can be represented approximately as a cylinder 1.6 \(\mathrm{m}\) long and 0.10 \(\mathrm{m}\) in diameter. The skin resistance can be made negligible by soaking the hands in salt water. (a) What is the resistance between the hands if the skin resistance is negligible? (b) What potential difference between the hands is needed for a lethal shock current of 100 \(\mathrm{mA}\) ? (Note that your result shows that small potential differences produce dangerous currents when the skin is damp.) (c) With the current in part (b), what power is dissipated in the body?

Calculate the (a) maximum and (b) minimum values of resistance that can be obtained by combining resistors of \(36 \Omega,\) \(47 \Omega,\) and 51\(\Omega .\)

A 40.0\(\Omega\) resistor and a 90.0\(\Omega\) resistor are connected in parallel, and the combination is connected across a \(120-\mathrm{V}\) dc line. (a) What is the resistance of the parallel combination? (b) What is the total current through the parallel combination? (c) What is the current through each resistor?

Three resistors having resistances of \(1.60 \Omega, 2.40 \Omega,\) and \(4.80 \Omega,\) respectively, are connected in parallel to a 28.0 \(\mathrm{V}\) bat- tery that has negligible internal resistance. Find (a) the equivalent resistance of the combination, (b) the current in each resistor, (c) the total current through the battery, (d) the voltage across each resistor, and (e) the power dissipated in each resistor. (f) Which resistor dissipates the most power, the one with the greatest resistance or the one with the least resistance? Explain why this should be.

Lightbulbs in series, I. The power rating of a lightbulb is the power it consumes when connected across a 120 \(\mathrm{V}\) out-let. (a) If you put two 100 \(\mathrm{W}\) bulbs in series across a 120 \(\mathrm{V}\) outlet, how much power would each consume if its resistance were constant? (b) How much power does each one consume if you connect them in parallel across a 120 \(\mathrm{V}\) outlet?

You absentmindedly solder a 69.8 \(\mathrm{k} \Omega\) resistor into a circuit where a 36.5 \(\mathrm{k} \Omega\) should be. How can you get the proper resistance without replacing the bigger resistor or removing anything from the circuit?

You need to connect a 68 \(\mathrm{k} \Omega\) resistor and one other resistor to a 110 \(\mathrm{V}\) power line. If you want the two resistors to use 4 times as much power when connected in parallel as they use when connected in series, what should be the value of the unknown resistor?

A 500.0\(\Omega\) resistor is connected in series with a capacitor. What must be the capacitance of the capacitor to produce a time constant of 2.00 \(\mathrm{s} ?\)

A fully charged 6.0\(\mu\) F capacitor is connected in series with a \(1.5 \times 10^{5} \Omega\) resistor. What percentage of the original charge is left on the capacitor after 1.8 s of discharging?

A 12.4 \(\mu \mathrm{F}\) capacitor is connected through a 0.895 \(\mathrm{M\Omega}\) resistor to a constant potential difference of 60.0 \(\mathrm{V}\) . (a) Compute the charge on the capacitor at the following times after the connections are made: \(0,5.0 \mathrm{s}, 10.0 \mathrm{s}, 20.0 \mathrm{s},\) and 100.0 \(\mathrm{s}\) . (b) Compute the charging currents at the same instants. (c) Graph the results of parts (a) and (b) for \(t\) between 0 and 20 \(\mathrm{s}\) .

A 6.00 \(\mu \mathrm{F}\) capacitor that is initially uncharged is connected in series with a 4500\(\Omega\) resistor and a 500 \(\mathrm{V}\) emf source with negligible internal resistance. Just after the circuit is completed, what are (a) the voltage drop across the capacitor, (b) the voltage drop across the resistor, (c) the charge on the capacitor, and (d) the current through the resistor? (e) A long time after the circuit is completed (after many time constants), what are the values of the preceding four quantities?

A capacitor is charged to a potential of 12.0 \(\mathrm{V}\) and is then connected to a voltmeter having an internal resistance of 3.40 \(\mathrm{M\Omega} .\) After a time of 4.00 \(\mathrm{s}\) the voltmeter reads 3.0 \(\mathrm{V}\) What are (a) the capacitance and (b) the time constant of the circuit?

Charging and discharging a capacitor. A 1.50\(\mu \mathrm{F}\) capacitor is charged through a 125\(\Omega\) resistor and then discharged through the same resistor by short-circuiting the battery. While the capacitor is being charged, find (a) the time for the charge on its plates to reach \(1-1 / e\) of its maximum value and (b) the current in the circuit at that time. (c) During the discharge of the capacitor, find the time for the charge on its plates to decrease to 1/e of its initial value. Also, find the time for the current in the circuit to decrease to 1\(/ e\) of its initial value.

Charging and discharging a capacitor. An initially uncharged capacitor \(C\) charges through a resistor \(R\) for many time constants and then discharges through the same resistor. Call \(Q_{\max }\) the maximum charge on its plates and \(I_{\max }\) the maximum current in the circuit. (a) Sketch clear graphs of the charge on the plates and the current in the circuit as functions of time for the charging process. (b) During the discharging process, the charge on the capacitor and the current both decrease exponentially from their maximum values. Use this fact to sketch graphs of the current in the circuit and the charge on the capacitor as functions of time.

A refrigerator draws 3.5 \(\mathrm{A}\) of current while operating on a 120 \(\mathrm{V}\) power line. If the refrigerator runs 50\(\%\) of the time and electric power costs \(\$ 0.12\) per kWh, how much does it cost to run this refrigerator for a 30 -day month?

A toaster using a Nichrome TM heating element operates on 120 \(\mathrm{V} .\) When it is switched on at \(20^{\circ} \mathrm{C},\) the heating element carries an initial current of 1.35 A. A few seconds later, the current reaches the steady value of 1.23 \(\mathrm{A}\) . (a) What is the final temperature of the element? The average value of the temperature coefficient of resistivity for Nichrome TM over the temperature range from \(20^{\circ} \mathrm{C}\) to the final temperature of the element is \(4.5 \times 10^{-4}\left(\mathrm{C}^{\circ}\right)^{-1} .\) (b) What is the power dissipated in the heating element (i) initially; (ii) when the current reaches a steady value?

A piece of wire has a resistance \(R\) . It is cut into three pieces of equal length, and the pieces are twisted together parallel to each other. What is the resistance of the resulting wire in terms of \(R ?\)

Flashlight batteries. A typical small flashlight contains two batteries, each having an emf of 1.5 \(\mathrm{V}\) , connected in series with a bulb having resistance 17\(\Omega .\) (a) If the internal resistance of the batteries is negligible, what power is delivered to the bulb? (b) If the batteries last for 5.0 h, what is the total energy delivered to the bulb? (c) The resistance of real batteries increases as they run down. If the initial internal resistance is negligible, what is the combined internal resistance of both batteries when the power to the bulb has decreased to half its initial value? (Assume that the resistance of the bulb is constant. Actually, it will change somewhat when the current through the filament changes, because this changes the temperature of the filament and hence the resistivity of the filament wire.)

Struck by lightning. Lightning strikes can involve currents as high as \(25,000\) A that last for about 40\(\mu\) s. If a person is struck by a bolt of lightning with these properties, the current will pass through his body. We shall assume that his mass is 75 kg, that he is wet (after all, he is in a rainstorm) and therefore has a resistance of \(1.0 \mathrm{k} \Omega,\) and that his body is all water (which is reasonable for a rough, but plausible, approximation).(a) By how many degrees Celsius would this lightning bolt increase the temperature of 75 kg of water? (b) Given that the internal body temperature is about \(37^{\circ} \mathrm{C}\) , would the person's temperature actually increase that much? Why not? What would happen first?