Solved: The starter motor of a car engine draws a current

What is current?

How does current flow?

What law governs current?

What are resistivity and resistance?

What is Ohms law?

What is the electron current in a 2.0-mm-diameter copper wire if the electron drift speed is 1.0 * 10-4 m/s?

These four wires are made of the same metal. Rank in order, from largest to smallest, the electron currents ia to id. (a) (b) (c) (d) 2r r 2v 2v

Why does the light in a room come on instantly when you flip a switch several meters away?

Table 23.1 in Chapter 23 gave a typical electric field strength in a current-carrying wire as 0.01 N/C or, as we would now say, 0.01 V/m. (Well verify this value later in this chapter.) Two 2.0-mm-diameter rings are 2.0 mm apart. They are charged to {Q. What value of Q causes the electric field at the midpoint to be 0.010 V/m?

The two charged rings are a model of the surface charge distribution along a wire. Rank in order, from largest to smallest, the electron currents Ea to Ee at the midpoint between the rings. (a) (b) (c) (d) (e)

Example 27.1 found the electron current to be 2.7 * 1019 s-1 for a 2.0-mm-diameter copper wire in which the electron drift speed is 1.0 * 10-4 m/s. If an internal electric field of 0.020 V/m is needed to sustain this current, a typical value, how many collisions per second, on average, do electrons in copper undergo?

A 1.0 A current passes through a 1.0-mm-diameter aluminum wire. What are the current density and the drift speed of the electrons in the wire?

What are the magnitude and the direction of the current in the fifth wire?

A 2.0-mm-diameter aluminum wire carries a current of 800 mA. What is the electric field strength inside the wire?

Rank in order, from largest to smallest, the current densities Ja to Jd in these four wires. (a) (b) (c) (d) r r 2r r I 2I 2I I s s 2s

Resistivity measurements on the leaves of corn plants are a good way to assess stress and the plants overall health. To determine resistivity, the current is measured when a voltage is applied between two electrodes placed 20 cm apart on a leaf that is 2.5 cm wide and 0.20 mm thick. The following data are obtained by using several different voltages: What is the resistivity of the leaf tissue?

A wire connects the positive and negative terminals of a battery. Two identical wires connect the positive and negative terminals of an identical battery. Rank in order, from largest to smallest, the currents Ia to Id at points a to d. E a b c Identical wires Identical batteries E d

What resistor would have a 15 mA current if connected across the terminals of a 9.0 V battery?

The woman in the photo on page 753 is gripping a device that measures body fat. To illustrate how this works, Figure 27.22 models an upper arm as part muscle and part fat, showing the resistivities of each. Nonconductive elements, such as skin and bone, have been ignored. This is obviously not a picture of the actual structure, but gathering all the fat tissue together and all the muscle tissue together is a model that predicts the arms electrical character quite well. A 0.87 mA current is recorded when a 0.60 V potential difference is applied across an upper arm having the dimensions shown in the figure. What are the percentages of muscle and fat in this persons upper arm?

Suppose a time machine has just brought you forward from 1750 (post-Newton but pre-electricity) and youve been shown the lightbulb demonstration of Figure Q27.1. Do observations or simple measurements you might makemeasurements that must make sense to you with your 1700s knowledgeprove that something is flowing through the wires? Or might you advance an alternative hypothesis for why the bulb is glowing? If your answer to the first question is yes, state what observations and/or measurements are relevant and the reasoning from which you can infer that something must be flowing. If not, can you offer an alternative hypothesis about why the bulb glows that could be tested?

Consider a lightbulb circuit such as the one in Figure Q27.1. a. From the simple observations and measurements you can make on this circuit, can you distinguish a current composed of positive charge carriers from a current composed of negative charge carriers? If so, describe how you can tell which it is. If not, why not? b. One model of current is the motion of discrete charged particles. Another model is that current is the flow of a continuous charged fluid. Do simple observations and measurements on this circuit provide evidence in favor of either one of these models? If so, describe how. If not, why not?

The electron drift speed in a wire is exceedingly slowtypically only a fraction of a millimeter per second. Yet when you turn on a flashlight switch, the light comes on almost instantly. Resolve this apparent paradox.

Is Figure Q27.4 a possible surface charge distribution for a current-carrying wire? If so, in which direction is the current? If not, why not?

What is the difference between current and current density?

All the wires in Figure Q27.6 are made of the same material and have the same diameter. Rank in order, from largest to smallest, the currents Ia to Id. Explain.

Both batteries in Figure Q27.7 are ideal and identical, and all lightbulbs are the same. Rank in order, from brightest to least bright, the brightness of bulbs a to c. Explain.

Both batteries in Figure Q27.8 are ideal and identical, and all lightbulbs are the same. Rank in order, from brightest to least bright, the brightness of bulbs a to c. Explain.

The wire in Figure Q27.9 consists of two segments of different diameters but made from the same metal. The current in segment 1 is I1. a. Compare the currents in the two segments. That is, is I2 greater than, less than, or equal to I1? Explain. b. Compare the current densities J1 and J2 in the two segments. c. Compare the electric field strengths E1 and E2 in the two segments. d. Compare the drift speeds (vd)1 and (vd)2 in the two segments

The current in a wire is doubled. What happens to (a) the current density, (b) the conduction-electron density, (c) the mean time between collisions, and (d) the electron drift speed? Are each of these doubled, halved, or unchanged? Explain.

The wires in Figure Q27.11 are all made of the same material. Rank in order, from largest to smallest, the resistances Ra to Re of these wires. Explain. I1 1 2 Figure Q27.9 2r 4L a b c d e L r L 2r r 2L 2

Which, if any, of these statements are true? (More than one may be true.) Explain. Assume the batteries are ideal. a. A battery supplies the energy to a circuit. b. A battery is a source of potential difference; the potential difference between the terminals of the battery is always the same. c. A battery is a source of current; the current leaving the battery is always the same.

1.0 * 1020 electrons flow through a cross section of a 2.0-mmdiameter iron wire in 5.0 s. What is the electron drift speed?

The electron drift speed in a 1.0-mm-diameter gold wire is 5.0 * 10-5 m/s. How long does it take 1 mole of electrons to flow through a cross section of the wire?

1.0 * 1016 electrons flow through a cross section of a silver wire in 320 ms with a drift speed of 8.0 * 10-4 m/s. What is the diameter of the wire?

Electrons flow through a 1.6-mm-diameter aluminum wire at 2.0 * 10-4 m/s. How many electrons move through a cross section of the wire each day?

The electron drift speed is 2.0 * 10-4 m/s in a metal with a mean time between collisions of 5.0 * 10-14 s. What is the electric field strength?

a. How many conduction electrons are there in a 1.0-mmdiameter gold wire that is 10 cm long? b. How far must the sea of electrons in the wire move to deliver -32 nC of charge to an electrode?

A 2.0 * 10-3 V/m electric field creates a 3.5 * 1017 electrons/s current in a 1.0-mm-diameter aluminum wire. What are (a) the drift speed and (b) the mean time between collisions for electrons in this wire?

The mean time between collisions in iron is 4.2*10-15 s. What electron current is driven through a 1.8-mm-diameter iron wire by a 0.065 V/m electric field?

The wires leading to and from a 0.12-mm-diameter lightbulb filament are 1.5 mm in diameter. The wire to the filament carries a current with a current density of 4.5 * 105 A/m2 . What are (a) the current and (b) the current density in the filament?

The current in a 100 watt lightbulb is 0.85 A. The filament inside the bulb is 0.25 mm in diameter. a. What is the current density in the filament? b. What is the electron current in the filament?

In an integrated circuit, the current density in a 2.5@mm@thick * 75@mm@wide gold film is 7.5 * 105 A/m2 . How much charge flows through the film in 15 min?

The current in an electric hair dryer is 10.0 A. How many electrons flow through the hair dryer in 5.0 min?

When a nerve cell fires, charge is transferred across the cell membrane to change the cells potential from negative to positive. For a typical nerve cell, 9.0 pC of charge flows in a time of 0.50 ms. What is the average current through the cell membrane?

The current in a 2.0 mm * 2.0 mm square aluminum wire is 2.5 A. What are (a) the current density and (b) the electron drift speed?

A hollow copper wire with an inner diameter of 1.0 mm and an outer diameter of 2.0 mm carries a current of 10 A. What is the current density in the wire?

A car battery is rated at 90 Ah, meaning that it can supply a 90 A current for 1 h before being completely discharged. If you leave your headlights on until the battery is completely dead, how much charge leaves the battery?

What is the mean time between collisions for electrons in (a) an aluminum wire and (b) an iron wire?

The electric field in a 2.0 mm * 2.0 mm square aluminum wire is 0.012 V/m. What is the current in the wire?

A 15-cm-long nichrome wire is connected across the terminals of a 1.5 V battery. a. What is the electric field inside the wire? b. What is the current density inside the wire? c. If the current in the wire is 2.0 A, what is the wires diameter?

What electric field strength is needed to create a 5.0 A current in a 2.0-mm-diameter iron wire?

A 0.0075 V/m electric field creates a 3.9 mA current in a 1.0-mm-diameter wire. What material is the wire made of?

A 3.0-mm-diameter wire carries a 12 A current when the electric field is 0.085 V/m. What is the wires resistivity?

A 0.50-mm-diameter silver wire carries a 20 mA current. What are (a) the electric field and (b) the electron drift speed in the wire?

The two segments of the wire in Figure EX27.24 have equal diameters but different conductivities s1 and s2. Current I passes through this wire. If the conductivities have the ratio s2/s1 = 2, what is the ratio E2/E1 of the electric field strengths in the two segments of the wire?

A 1.5 V battery provides 0.50 A of current. a. At what rate (C/s) is charge lifted by the charge escalator? b. How much work does the charge escalator do to lift 1.0 C of charge? c. What is the power output of the charge escalator?

Wires 1 and 2 are made of the same metal. Wire 2 has twice the length and twice the diameter of wire 1. What are the ratios (a) r2/r1 of the resistivities and (b) R2/R1 of the resistances of the two wires?

What is the resistance of a. A 2.0-m-long gold wire that is 0.20 mm in diameter? b. A 10-cm-long piece of carbon with a 1.0 mm * 1.0 mm square cross section?

An engineer cuts a 1.0-m-long, 0.33-mm-diameter piece of wire, connects it across a 1.5 V battery, and finds that the current in the wire is 8.0 A. Of what material is the wire made?

The electric field inside a 30-cm-long copper wire is 5.0 mV/m. What is the potential difference between the ends of the wire?

a. How long must a 0.60-mm-diameter aluminum wire be to have a 0.50 A current when connected to the terminals of a 1.5 V flashlight battery? b. What is the current if the wire is half this length?

The terminals of a 0.70 V watch battery are connected by a 100-m-long gold wire with a diameter of 0.10 mm. What is the current in the wire?

The femoral artery is the large artery that carries blood to the leg. What is the resistance of a 20-cm-long column of blood in a 1.0-cm-diameter femoral artery? The conductivity of blood is 0.63 -1m-1 .

Pencil lead is actually carbon. What is the current if a 9.0 V potential difference is applied between the ends of a 0.70-mmdiameter, 6.0-cm-long lead from a mechanical pencil?

The resistance of a very fine aluminum wire with a 10 mm * 10 mm square cross section is 1000 . A 1000 resistor is made by wrapping this wire in a spiral around a 3.0-mm-diameter glass core. How many turns of wire are needed?

Figure EX27.35 is a currentversus-potential-difference graph for a material. What is the materials resistance?

A circuit calls for a 0.50-mm-diameter copper wire to be stretched between two points. You dont have any copper wire, but you do have aluminum wire in a wide variety of diameters. What diameter aluminum wire will provide the same resistance?

Household wiring often uses 2.0-mm-diameter copper wires. The wires can get rather long as they snake through the walls from the fuse box to the farthest corners of your house. What is the potential difference across a 20-m-long, 2.0-mm-diameter copper wire carrying an 8.0 A current?

An ideal battery would produce an extraordinarily large current if shorted by connecting the positive and negative terminals with a short wire of very low resistance. Real batteries do not. The current of a real battery is limited by the fact that the battery itself has resistance. What is the resistance of a 9.0 V battery that produces a 21 A current when shorted by a wire of negligible resistance?

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 3.5 mA. How many protons are striking the detector per second?

Figure P27.42 shows a 4.0-cmwide 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 nC/cm2 . a. What is the current of the moving film? b. How long does it take the roller to accumulate a charge of -10 mC?

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 youve 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?

In a classic model of the hydrogen atom, the electron moves around the proton in a circular orbit of radius 0.053 nm. a. What is the electrons orbital frequency? b. What is the effective current of the electron?

Youve 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 mm * 1.0 mm * 45 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: Voltage (V) Current (A) 0.200 0.47 0.400 1.06 0.600 1.53 0.800 1.97 Use an appropriate graph of the data to determine whether the material is ohmic and, if so, its conductivity.

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 1K+2 channel carries a current of 1.8 pA. a. How many potassium ions pass through if the ion channel opens for 1.0 ms? b. What is the current density in the ion channel?

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. a. How much charge passes through the starter motor? b. How far does an electron travel along the wire while the starter motor is on?

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 resistivity of a metal increases slightly with increased temperature. This can be expressed as r = r0 31 + a1T - T024, where T0 is a reference temperature, usually 20C, and a is the temperature coefficient of resistivity. For copper, a = 3.9 * 10-3 C-1 . Suppose a long, thin copper wire has a resistance of 0.25 at 20C. At what temperature, in C, will its resistance be 0.30 ?

Variations in the resistivity of blood can give valuable clues about changes in various properties of the blood. Suppose a medical device attaches two electrodes 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 mA current through the blood in the vein?

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 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

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 m and 25 m, respectively. One persons upper leg is 82% muscle, 18% fat. What current is measured if a 1.5 V potential difference is applied between the persons hip and knee?

The resistivity of a metal increases slightly with increased temperature. This can be expressed as r = r0 31 + a1T - T024, where T0 is a reference temperature, usually 20C, and a is the temperature coefficient of resistivity. a. 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 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 a in the numerator. b. For copper, a = 3.9 * 10-3 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 20C? c. What is the rate, in A/C, at which the current changes with temperature as the wire heats up?

Electrical engineers sometimes use a wires conductance, G = sA/L, instead of its resistance. a. Write Ohms law in terms of conductance, starting with V = . b. What is the conductance of a 5.4-cm-long, 0.15-mm-diameter tungsten wire? c. 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 A/cm2 . What diameter wire of this material will do the job?

A hollow metal cylinder has inner radius a, outer radius b, length L, and conductivity s. The current I is radially outward from the inner surface to the outer surface. a. Find an expression for the electric field strength inside the metal as a function of the radius r from the cylinders axis. b. Evaluate the electric field strength at the inner and outer surfaces of an iron cylinder if a = 1.0 cm, b = 2.5 cm, L = 10 cm, and I = 25 A.

A hollow metal sphere has inner radius a, outer radius b, and conductivity s. The current I is radially outward from the inner surface to the outer surface. a. Find an expression for the electric field strength inside the metal as a function of the radius r from the center. b. Evaluate the electric field strength at the inner and outer surfaces of a copper sphere if a = 1.0 cm, b = 2.5 cm, and I = 25 A

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 0. a. Find an expression for the current in the wire at time t. b. Graph I versus t for the interval 0 t 4 s

The total amount of charge that has entered a wire at time t is given by the expression Q = 120 C211 - e-t/12.0 s2 2, where t is in seconds and t 0. a. Find an expression for the current in the wire at time t. b. What is the maximum value of the current? c. Graph I versus t for the interval 0 t 10 s

The current in a wire at time t is given by the expression I = 12.0 A2e-t/12.0 ms2 , where t is in microseconds and t 0. a. Find an expression for the total amount of charge (in coulombs) that has entered the wire at time t. The initial conditions are Q = 0 C at t = 0 ms. b. Graph Q versus t for the interval 0 t 10 ms

The current supplied by a battery slowly decreases as the battery runs down. Suppose that the current as a function of time is I = 10.75 A2e-t/16 h2 . 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?

The two wires in FIGURE P27.62 are made of the same material. What are the current and the electron drift speed in the 2.0-mmdiameter segment of the wire?

What diameter should the nichrome wire in FIGURE P27.63 be in order for the electric field strength to be the same in both wires?

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 a. Current I. b. Current density J. c. Electric field E. d. Drift velocity vd. e. Electron current i. Place your results in a table for easy viewing

A wire of radius R has a current density that increases linearly with distance from the center of the wire: J1r2 = kr, where k is a constant. Find an expression for k in terms of R and the total current I carried by the wire

A 0.60-mm-diameter wire made from an alloy (a combination of different metals) has a conductivity that decreases linearly with distance from the center of the wire: s1r2 = s0 - cr, with s0 = 5.0 * 107 -1m-1 and c = 1.2 * 1011 -1m-2 . What is the resistance of a 4.0 m 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?

A 1.5 V flashlight battery is connected to a wire with a resistance of 3.0 . Figure P27.69 shows the batterys potential difference as a function of time. What is the total charge lifted by the charge escalator?

Two 10-cm-diameter metal plates 1.0 cm apart are charged to {12.5 nC. They are suddenly connected together by a 0.224-mmdiameter copper wire stretched taut from the center of one plate to the center of the other. a. What is the maximum current in the wire? b. Does the current increase with time, decrease with time, or remain steady? Explain. c. What is the total amount of energy dissipated in the wire?

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

Youve 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, youve learned that lightning bolts can carry up to 50 kA of current and last up to 50 ms.a. How much charge is delivered by a lightning bolt with these parameters? b. You dont want the potential difference between the top and bottom of the lightning rod to exceed 100 V. What minimum diameter must the rod have?

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 = Jedge1r/R2, where R is the radius of the beam and Jedge is the current density at the edge. a. How many protons per second are delivered by this proton beam? b. Determine the value of Jedge

Figure CP27.74 shows a wire that is made of two equaldiameter segments with conductivities s1 and s2. When current I passes through the wire, a thin layer of charge appears at the boundary between the segments. a. Find an expression for the surface charge density h on the boundary. Give your result in terms of I, s1, s2, and the wires cross-section area A. b. 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?

A 300 mF capacitor is charged to 9.0 V, then connected in parallel with a 5000 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, youll need a minus sign that you might not have expected

A thin metal cylinder of length L and radius R1 is coaxial with a thin metal cylinder of length L and a larger radius R2 . The space between the two coaxial cylinders is filled with a material that has resistivity r. The two cylinders are connected to the terminals of a battery with potential difference V, causing current I to flow radially from the inner cylinder to the outer cylinder. Find an expression for the resistance of this device.