|| Energetic particles, such as protons, can be detected with a silicon detector. When a particle strikes a thin piece of silicon, it creates a large number of free electrons by ionizing silicon atoms. The electrons flow to an electrode on the surface of the detector, and this current is then amplified and detected. In one experiment, each incident proton creates, on average, 35,000 electrons; the electron current is amplified by a factor of 100; and the experimenters record an amplified current of $3\text{.}5\mu \text{A}$. How many protons are striking the detector per second?

|| For what electric field strength would the current in a 2.0-mm-diameter nichrome wire be the same as the current in a 1.0-mm-diameter aluminum wire in which the electric field strength is $0.0080\text{V}\text{/}\text{m}\text{?}$

|| INT The electron beam inside an old television picture tube is 0.40 mm in diameter and carries a current of $50\mu \text{A}$. This electron beam impinges on the inside of the picture tube screen.

1. How many electrons strike the screen each second?

2. What is the current density in the electron beam?

3. The electrons move with a velocity of $4.0\times {10}^7\text{m}\text{/}\text{s}$. What electric field strength is needed to accelerate electrons from rest to this velocity in a distance of 5.0 mm?

4. Each electron transfers its kinetic energy to the picture tube screen upon impact. What is the power delivered to the screen by the electron beam?

|| INT A sculptor has asked you to help electroplate gold onto a brass statue. You know that the charge carriers in the ionic solution are singly charged gold ions, and you’ve calculated that you must deposit 0.50 g of gold to reach the necessary thickness. How much current do you need, in mA, to plate the statue in 3.0 hours?

|| INT FIGURE P27.43 shows a 4.0-cm-wide plastic film being wrapped onto a 2.0-cm-diameter roller that turns at 90 rpm. The plastic has a uniform surface charge density of $-2.0\text{nC}\text{/}\text{c}{\text{m}}^2$.

1. What is the current of the moving film?

2. How long does it take the roller to accumulate a charge of $-10\mu \text{C}\text{?}$

FIGURE P27.43

|| Thermistors, resistors whose resistance is a sensitive function of temperature, are widely used in industry and consumer devices to measure temperature. The resistance of a thermistor at temperature $T$ can be modeled as $R=R_0\text{exp}\left[\beta (1/T-1/T_0)\right]$, where $T_0$ is a reference temperature, the temperatures are in K, and $\beta$ is a constant with units of K. Suppose you connect a thermistor to a 10.0 V battery and measure the current through it at different temperatures. At 25.0°C, which you select as your reference temperature, the current is 10.0 mA.

1. What is the value of $R_0\text{?}$

2. Raising the temperature to 30.0°C causes the current to increase to 12.5 mA. What is the value of $\beta \text{?}$

3. What is the temperature in °C when the current is 5.0 mA?

|| You’ve been asked to determine whether a new material your company has made is ohmic and, if so, to measure its electrical conductivity. Taking a $0.50\text{mm}\times 1.0\text{mm}\times 45\text{mm}$ sample, you wire the ends of the long axis to a power supply and then measure the current for several different potential differences. Your data are as follows:

Use an appropriate graph of the data to determine whether the material is ohmic and, if so, its conductivity.

| BIO The biochemistry that takes place inside cells depends on various elements, such as sodium, potassium, and calcium, that are dissolved in water as ions. These ions enter cells through narrow pores in the cell membrane known as ion channels. Each ion channel, which is formed from a specialized protein molecule, is selective for one type of ion. Measurements with microelectrodes have shown that a 0.30-nm-diameter potassium ion $(\text{K}{}^{+})$ channel carries a current of 1.8 pA.

1. How many potassium ions pass through if the ion channel opens for 1.0 ms?

2. What is the current density in the ion channel?

|| A 1.5-m-long wire is made of a metal with the same electron density as copper. The wire is connected across the terminals of a 9.0 V battery. What conductivity would the metal need for the drift velocity of electrons in the wire to be 60 mph? By what factor is this larger than the conductivity of copper?

| The starter motor of a car engine draws a current of 150 A from the battery. The copper wire to the motor is 5.0 mm in diameter and 1.2 m long. The starter motor runs for 0.80 s until the car engine starts.

1. How much charge passes through the starter motor?

2. How far does an electron travel along the wire while the starter motor is on?

|| The resistivity of a metal increases slightly with increased temperature. This can be expressed as $\rho ={\rho }_0[{}_{}1+\alpha (T-T_0)]$, where $T_0$ is a reference temperature, usually 20°C, and $\alpha$ is the temperature coefficient of resistivity. For copper, $\alpha =3\text{.}9\times {10}^{-3}\text{°}{\text{C}}^{-1}$. Suppose a long, thin copper wire has a resistance of $0\text{.}25\text{Ω}$ at 20°C. At what temperature, in °C, will its resistance be $0\text{.}30\text{Ω}\text{?}$

|| BIO Variations in the resistivity of blood can give valuable clues about changes in various properties of the blood. Suppose a medical device inserts microelectrodes into a 1.5-mm-diameter vein at positions 5.0 cm apart. What is the blood resistivity if a 9.0 V potential difference causes a $230\mu \text{A}$ current through the blood in the vein?

|| BIO The conducting path between the right hand and the left hand can be modeled as a 10-cm-diameter, 160-cm-long cylinder. The average resistivity of the interior of the human body is $5.0\text{Ω}\text{m}$. Dry skin has a much higher resistivity, but skin resistance can be made negligible by soaking the hands in salt water. If skin resistance is neglected, what potential difference between the hands is needed for a lethal shock of 100 mA across the chest? Your result shows that even small potential differences can produce dangerous currents when the skin is wet.

||| BIO The conductive tissues of the upper leg can be modeled as a 40-cm-long, 12-cm-diameter cylinder of muscle and fat. The resistivities of muscle and fat are $13\text{Ω}\text{m}$ and $25\text{Ω}\text{m,}$ respectively. One person’s upper leg is 82% muscle, 18% fat. What current is measured if a 1.5 V potential difference is applied between the person’s hip and knee?

|| Electrical engineers sometimes use a wire’s conductance, $G=\sigma A\text{/}L$, instead of its resistance.

1. Write Ohm’s law in terms of conductance, starting with “$\text{Δ}V=$”.

2. What is the conductance of a 5.4-cm-long, 0.15-mm-diameter tungsten wire?

3. A 1.5 A current flows through the wire of part b. What is the potential difference between the ends of the wire?

|| You need to design a 1.0 A fuse that “blows” if the current exceeds 1.0 A. The fuse material in your stockroom melts at a current density of $500\text{A}\text{/}\text{c}{\text{m}}^2$. What diameter wire of this material will do the job?

|| A superconducting wire has a critical current density at which the wire suddenly loses its superconductivity and becomes a normal conductor. This is called a quench of superconductivity, and it’s potentially very damaging for a superconducting magnet. A superconducting magnet you’re designing will wind a coil with a 1.0-mm-diameter wire whose critical current density is $950{\text{A/mm}}^2$. To have a margin of safety, you need to limit the maximum possible current density to 80% of the critical value. What is the maximum allowable current in the wire?

|| A hollow metal cylinder has inner radius $a$, outer radius $b$, length $L$, and conductivity $\sigma$. The current $I$ is radially outward from the inner surface to the outer surface.

1. Find an expression for the electric field strength inside the metal as a function of the radius $r$ from the cylinder’s axis.

2. Evaluate the electric field strength at the inner and outer surfaces of an iron cylinder if $a=1.0\text{cm,}$ $b=2.5\text{cm,}$ $L=10\text{cm,}$ and $I=25\text{A}$.

|| CALC The resistivity of a metal increases slightly with increased temperature. This can be expressed as $\rho ={\rho }_0[1+\alpha (T-T_0)]$, where $T_0$ is a reference temperature, usually 20°C, and $\alpha$ is the temperature coefficient of resistivity.

1. First find an expression for the current $I$ through a wire of length $L$, cross-section area $A$, and temperature $T$ when connected across the terminals of an ideal battery with terminal voltage $\text{Δ}V$. Then, because the change in resistance is small, use the binomial approximation to simplify your expression. Your final expression should have the temperature coefficient $\alpha$ in the numerator.

2. For copper, $\alpha =3\text{.}9\times {10}^{-3}\text{°}{\text{C}}^{-1}$. Suppose a 2.5-m-long, 0.40-mm-diameter copper wire is connected across the terminals of a 1.5 V ideal battery. What is the current in the wire at 20°C?

3. What is the rate, in $\text{A}\text{/}\text{°}\text{C}$, at which the current changes with temperature as the wire heats up?

|| CALC The total amount of charge in coulombs that has entered a wire at time $t$ is given by the expression $Q=4t-t^2$, where $t$ is in seconds and $t\ge 0$.

1. Find an expression for the current in the wire at time $t$.

2. Graph $I$ versus $t$ for the interval $0\le t\le 4\text{s}$.

|| CALC The total amount of charge that has entered a wire at time $t$ is given by the expression $Q=(20\text{C})(1-e^{-t\text{/}(2.0\text{s})})$, where $t$ is in seconds and $t\ge 0$.

1. Find an expression for the current in the wire at time $t$.

2. What is the maximum value of the current?

3. Graph $I$ versus $t$ for the interval $0\le t\le 10\text{s}$.

|| CALC The current in a wire at time $t$ is given by the expression $I=(2.0\text{A})e^{-t\text{/}(2.0\mu \text{s})}$, where $t$ is in microseconds and $t\ge 0$.

1. Find an expression for the amount of charge (in coulombs) that has entered the wire at time $t$. The initial conditions are $Q=0\text{C}$ at $t=0\mu \text{s}$.

2. Graph $Q$ versus $t$ for the interval $0\le t\le 10\mu \text{s}$.

|| CALC The current supplied by a battery slowly decreases as the battery runs down. Suppose that the current as a function of time is $I=(0.75\text{A})e^{-t\text{/}(6\text{h})}$. What is the total number of electrons transported from the positive electrode to the negative electrode by the charge escalator from the time the battery is first used until it is completely dead?

| What diameter should the nichrome wire in FIGURE P27.62 be in order for the electric field strength to be the same in both wires?

FIGURE P27.62

| The two wires in FIGURE P27.63 are made of the same material. What are (a) the current and (b) the electron drift speed in the 2.0-mm-diameter segment of the wire?

FIGURE P27.63

|| An aluminum wire consists of the three segments shown in FIGURE P27.64. The current in the top segment is 10 A. For each of these three segments, find the

1. Current $I$.

2. Current density $J$.

3. Electric field $E$.

4. Drift velocity $v_{\text{d}}$.

5. Electron current $i_{\text{e}}$.

Place your results in a table for easy viewing.

FIGURE P27.64

| You’ve decided to protect your house by placing a 5.0-m-tall iron lightning rod next to the house. The top is sharpened to a point and the bottom is in good contact with the ground. From your research, you’ve learned that lightning bolts can carry up to 50 kA of current and last up to $50\mu \text{s}$.

1. How much charge is delivered by a lightning bolt with these parameters?

2. You don’t want the potential difference between the top and bottom of the lightning rod to exceed 100 V. What minimum diameter must the rod have?

|| CALC A 2.0-mm-diameter wire formed from a composite material has a resistivity that decreases with distance along the wire as $\rho ={\rho }_0{e}^{-\alpha x}$, where ${\rho }_0=4.0\times {10}^{-5}\text{Ω}\text{m}$, $x$ (in m) is measured from one end of the wire, and the constant $\alpha =4.0\text{m}{}^{-1}$. What is the resistance of a 50-cm-long length of this wire?

|| A 20-cm-long hollow nichrome tube of inner diameter 2.8 mm, outer diameter 3.0 mm is connected, at its ends, to a 3.0 V battery. What is the current in the tube?

| The batteries in FIGURE P27.68 are identical. Both resistors have equal currents. What is the resistance of the resistor on the right?

FIGURE P27.68

|| A 1.5 V flashlight battery is connected to a wire with a resistance of $3.0\text{Ω}$. FIGURE P27.69 shows the battery’s potential difference as a function of time. What is the total charge lifted by the charge escalator?

FIGURE P27.69

|| INT Two 10-cm-diameter metal plates 1.0 cm apart are charged to $\pm 12.5\text{nC}$. They are suddenly connected together by a 0.224-mm-diameter copper wire stretched taut from the center of one plate to the center of the other.

1. What is the maximum current in the wire?

2. Does the current increase with time, decrease with time, or remain steady? Explain.

3. What is the total amount of energy dissipated in the wire?

|| A long, round wire has resistance R. What will the wire’s resistance be if you stretch it to twice its initial length?

||| CALC A 5.0-mm-diameter proton beam carries a total current of 1.5 mA. The current density in the proton beam, which increases with distance from the center, is given by $J=J_{\text{edge}}\left(r\text{/}R\right)$, where $R$ is the radius of the beam and $J_{\text{edge}}$ is the current density at the edge.

1. How many protons per second are delivered by this proton beam?

2. Determine the value of $J_{\text{edge}}$.

||| CALC A wire of radius $R$ has a current density that increases linearly with distance from the center of the wire: $J\left(r\right)=kr$, where $k$ is a constant. Find an expression for $k$ in terms of $R$ and the total current $I$ carried by the wire.

||| FIGURE CP27.74 shows a wire that is made of two equal-diameter segments with conductivities ${\sigma }_1$ and ${\sigma }_2$. When current $I$ passes through the wire, a thin layer of charge appears at the boundary between the segments.

1. Find an expression for the surface charge density $\eta$ on the boundary. Give your result in terms of $I$, ${\sigma }_1$, ${\sigma }_2$, and the wire’s cross-section area $A$.

2. A 1.0-mm-diameter wire made of copper and iron segments carries a 5.0 A current. How much charge accumulates at the boundary between the segments?

FIGURE CP27.74

||| CALC A $300\mu \text{F}$ capacitor is charged to 9.0 V, then connected in parallel with a $5000\text{Ω}$ resistor. The capacitor will discharge because the resistor provides a conducting pathway between the capacitor plates, but much more slowly than if the plates were connected by a wire. Let $t=0$ s be the instant the fully charged capacitor is first connected to the resistor. At what time has the capacitor voltage decreased by half, to 4.5 V?

Hint: The current through the resistor is related to the rate at which charge is leaving the capacitor. Consequently, you’ll need a minus sign that you might not have expected.

||| CALC A thin metal cylinder of length $L$ and radius $R_1$ is coaxial with a thin metal cylinder of length $L$ and a larger radius $R_2$. The space between the two coaxial cylinders is filled with a material that has resistivity $\rho$. The two cylinders are connected to the terminals of a battery with potential difference $\text{Δ}V$, causing current $I$ to flow radially from the inner cylinder to the outer cylinder. Find an expression for the resistance of this device.