### Problems

### 9.1 Electrical Current

A Van de Graaff generator is one of the original particle accelerators and can be used to accelerate charged particles like protons or electrons. You may have seen it used to make human hair stand on end or produce large sparks. One application of the Van de Graaff generator is to create X-rays by bombarding a hard metal target with the beam. Consider a beam of protons at 1.00 keV and a current of 5.00 mA produced by the generator. (a) What is the speed of the protons? (b) How many protons are produced each second?

A cathode ray tube (CRT) is a device that produces a focused beam of electrons in a vacuum. The electrons strike a phosphor-coated glass screen at the end of the tube, which produces a bright spot of light. The position of the bright spot of light on the screen can be adjusted by deflecting the electrons with electrical fields, magnetic fields, or both. Although the CRT tube was once commonly found in televisions, computer displays, and oscilloscopes, newer appliances use a liquid crystal display (LCD) or plasma screen. You still may come across a CRT in your study of science. Consider a CRT with an electron beam average current of $25.00\mu \phantom{\rule{0.2em}{0ex}}\text{A}$. How many electrons strike the screen every minute?

How many electrons flow through a point in a wire in 3.00 s if there is a constant current of $I=4.00\phantom{\rule{0.2em}{0ex}}\text{A}$?

A conductor carries a current that is decreasing exponentially with time. The current is modeled as $I={I}_{0}{e}^{\text{\u2212}t\text{/}\tau}$, where ${I}_{0}=3.00\phantom{\rule{0.2em}{0ex}}\text{A}$ is the current at time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ and $\tau =0.50\phantom{\rule{0.2em}{0ex}}\text{s}$ is the time constant. How much charge flows through the conductor between $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ and $t=3\tau $?

The quantity of charge through a conductor is modeled as $Q=4.00\frac{\text{C}}{{\text{s}}^{4}}{t}^{4}-1.00\frac{\text{C}}{\text{s}}t+6.00\phantom{\rule{0.2em}{0ex}}\text{mC}$.

What is the current at time $t=3.00\phantom{\rule{0.2em}{0ex}}\text{s}$?

The current through a conductor is modeled as $I\left(t\right)={I}_{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\left(2\pi \left[60\phantom{\rule{0.2em}{0ex}}\text{Hz}\right]t\right)$. Write an equation for the charge as a function of time.

The charge on a capacitor in a circuit is modeled as $Q\left(t\right)={Q}_{\text{max}}\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\left(\omega t+\varphi \right)$. What is the current through the circuit as a function of time?

### 9.2 Model of Conduction in Metals

An aluminum wire 1.628 mm in diameter (14-gauge) carries a current of 3.00 amps. (a) What is the absolute value of the charge density in the wire? (b) What is the drift velocity of the electrons? (c) What would be the drift velocity if the same gauge copper were used instead of aluminum? The density of copper is $8.96{\phantom{\rule{0.2em}{0ex}}\text{g/cm}}^{3}$ and the density of aluminum is $2.70{\phantom{\rule{0.2em}{0ex}}\text{g/cm}}^{3}$. The molar mass of aluminum is 26.98 g/mol and the molar mass of copper is 63.5 g/mol. Assume each atom of metal contributes one free electron.

The current of an electron beam has a measured current of $I=50.00\phantom{\rule{0.2em}{0ex}}\mu \text{A}$ with a radius of 1.00 mm. What is the magnitude of the current density of the beam?

A high-energy proton accelerator produces a proton beam with a radius of $r=0.90\phantom{\rule{0.2em}{0ex}}\text{mm}$. The beam current is $I=9.00\phantom{\rule{0.2em}{0ex}}\mu \text{A}$ and is constant. The charge density of the beam is $n=6.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{11}$ protons per cubic meter. (a) What is the current density of the beam? (b) What is the drift velocity of the beam? (c) How much time does it take for $1.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{10}$ protons to be emitted by the accelerator?

Consider a wire of a circular cross-section with a radius of $R=3.00\phantom{\rule{0.2em}{0ex}}\text{mm}$. The magnitude of the current density is modeled as $J=c{r}^{2}=5.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{6}\frac{\text{A}}{{\text{m}}^{4}}{r}^{2}$. What is the current through the inner section of the wire from the center to $r=0.5R$?

A cylindrical wire has a current density from the center of the wire’s cross section as $J\left(r\right)=C{r}^{2}$ where $r$ is in meters, $J$ is in amps per square meter, and $C={10}^{3}\phantom{\rule{0.2em}{0ex}}{\text{A/m}}^{4}$. This current density continues to the end of the wire at a radius of 1.0 mm. Calculate the current just outside of this wire.

The current supplied to an air conditioner unit is 4.00 amps. The air conditioner is wired using a 10-gauge (diameter 2.588 mm) wire. The charge density is $n=8.48\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{28}\frac{\text{electrons}}{{\text{m}}^{3}}$. Find the magnitude of (a) current density and (b) the drift velocity.

### 9.3 Resistivity and Resistance

What current flows through the bulb of a 3.00-V flashlight when its hot resistance is $3.60\phantom{\rule{0.2em}{0ex}}\text{\Omega}$?

Calculate the effective resistance of a pocket calculator that has a 1.35-V battery and through which 0.200 mA flows.

How many volts are supplied to operate an indicator light on a DVD player that has a resistance of $140\phantom{\rule{0.2em}{0ex}}\text{\Omega}$, given that 25.0 mA passes through it?

What is the resistance of a 20.0-m-long piece of 12-gauge copper wire having a 2.053-mm diameter?

The diameter of 0-gauge copper wire is 8.252 mm. Find the resistance of a 1.00-km length of such wire used for power transmission.

If the 0.100-mm-diameter tungsten filament in a light bulb is to have a resistance of $0.200\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ at $20.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$, how long should it be?

A lead rod has a length of 30.00 cm and a resistance of $5.00\phantom{\rule{0.2em}{0ex}}\mu \text{\Omega}$. What is the radius of the rod?

Find the ratio of the diameter of aluminum to copper wire, if they have the same resistance per unit length (as they might in household wiring).

What current flows through a 2.54-cm-diameter rod of pure silicon that is 20.0 cm long, when $1.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{V}$ is applied to it? (Such a rod may be used to make nuclear-particle detectors, for example.)

(a) To what temperature must you raise a copper wire, originally at $20.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$, to double its resistance, neglecting any changes in dimensions? (b) Does this happen in household wiring under ordinary circumstances?

A resistor made of nichrome wire is used in an application where its resistance cannot change more than 1.00% from its value at $20.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$. Over what temperature range can it be used?

Of what material is a resistor made if its resistance is 40.0% greater at $100.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ than at $20.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$?

An electronic device designed to operate at any temperature in the range from $\mathrm{-10.0}\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ to $55.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ contains pure carbon resistors. By what factor does their resistance increase over this range?

(a) Of what material is a wire made, if it is 25.0 m long with a diameter of 0.100 mm and has a resistance of $77.7\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ at $20.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$? (b) What is its resistance at $150.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C?}$

Assuming a constant temperature coefficient of resistivity, what is the maximum percent decrease in the resistance of a constantan wire starting at $20.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$?

A copper wire has a resistance of $0.500\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ at $20.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C},$ and an iron wire has a resistance of $0.525\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ at the same temperature. At what temperature are their resistances equal?

### 9.4 Ohm's Law

A $2.2\text{-k}\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ resistor is connected across a D cell battery (1.5 V). What is the current through the resistor?

A resistor rated at $250\phantom{\rule{0.2em}{0ex}}\text{k}\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ is connected across two D cell batteries (each 1.50 V) in series, with a total voltage of 3.00 V. The manufacturer advertises that their resistors are within 5% of the rated value. What are the possible minimum current and maximum current through the resistor?

A resistor is connected in series with a power supply of 20.00 V. The current measure is 0.50 A. What is the resistance of the resistor?

A resistor is placed in a circuit with an adjustable voltage source. The voltage across and the current through the resistor and the measurements are shown below. Estimate the resistance of the resistor.

The following table show the measurements of a current through and the voltage across a sample of material. Plot the data, and assuming the object is an ohmic device, estimate the resistance.

I(A) |
V(V) |
---|---|

0 | 3 |

2 | 23 |

4 | 39 |

6 | 58 |

8 | 77 |

10 | 100 |

12 | 119 |

14 | 142 |

16 | 162 |

### 9.5 Electrical Energy and Power

A $20.00\text{-V}$ battery is used to supply current to a $10\text{-k}\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ resistor. Assume the voltage drop across any wires used for connections is negligible. (a) What is the current through the resistor? (b) What is the power dissipated by the resistor? (c) What is the power input from the battery, assuming all the electrical power is dissipated by the resistor? (d) What happens to the energy dissipated by the resistor?

What is the maximum voltage that can be applied to a $20\text{-k}\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ resistor rated at $\frac{1}{4}\text{W}$?

A heater is being designed that uses a coil of 14-gauge nichrome wire to generate 300 W using a voltage of $V=110\phantom{\rule{0.2em}{0ex}}\text{V}$. How long should the engineer make the wire?

An alternative to CFL bulbs and incandescent bulbs are light-emitting diode (LED) bulbs. A 100-W incandescent bulb can be replaced by a 16-W LED bulb. Both produce 1600 lumens of light. Assuming the cost of electricity is $0.10 per kilowatt-hour, how much does it cost to run the bulb for one year if it runs for four hours a day?

The power dissipated by a resistor with a resistance of $R=100\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ is $P=2.0\phantom{\rule{0.2em}{0ex}}\text{W}$. What are the current through and the voltage drop across the resistor?

Running late to catch a plane, a driver accidentally leaves the headlights on after parking the car in the airport parking lot. During takeoff, the driver realizes the mistake. Having just replaced the battery, the driver knows that the battery is a 12-V automobile battery, rated at 100 $\text{A}\xb7\text{h}$. The driver, knowing there is nothing that can be done, estimates how long the lights will shine, assuming there are two 12-V headlights, each rated at 40 W. What did the driver conclude?

A physics student has a single-occupancy dorm room. The student has a small refrigerator that runs with a current of 3.00 A and a voltage of 110 V, a lamp that contains a 100-W bulb, an overhead light with a 60-W bulb, and various other small devices adding up to 3.00 W. (a) Assuming the power plant that supplies 110 V electricity to the dorm is 10 km away and the two aluminum transmission cables use 0-gauge wire with a diameter of 8.252 mm, estimate the percentage of the total power supplied by the power company that is lost in the transmission. (b) What would be the result is the power company delivered the electric power at 110 kV?

A 0.50-W, $220\text{-}\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ resistor carries the maximum current possible without damaging the resistor. If the current were reduced to half the value, what would be the power consumed?

### 9.6 Superconductors

Consider a power plant is located 60 km away from a residential area uses 0-gauge $\left(A=42.40\phantom{\rule{0.2em}{0ex}}{\text{mm}}^{2}\right)$ wire of copper to transmit power at a current of $I=100.00\phantom{\rule{0.2em}{0ex}}\text{A}$. How much more power is dissipated in the copper wires than it would be in superconducting wires?

A wire is drawn through a die, stretching it to four times its original length. By what factor does its resistance increase?

Digital medical thermometers determine temperature by measuring the resistance of a semiconductor device called a thermistor (which has $\alpha =\mathrm{-0.06}\text{/}\text{\xb0}\text{C}$) when it is at the same temperature as the patient. What is a patient’s temperature if the thermistor’s resistance at that temperature is 82.0% of its value at $37\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ (normal body temperature)?

Electrical power generators are sometimes “load tested” by passing current through a large vat of water. A similar method can be used to test the heat output of a resistor. A $R=30\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ resistor is connected to a 9.0-V battery and the resistor leads are waterproofed and the resistor is placed in 1.0 kg of room temperature water $\left(T=20\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}\right)$. Current runs through the resistor for 20 minutes. Assuming all the electrical energy dissipated by the resistor is converted to heat, what is the final temperature of the water?

A 12-gauge gold wire has a length of 1 meter. (a) What would be the length of a silver 12-gauge wire with the same resistance? (b) What are their respective resistances at the temperature of boiling water?

What is the change in temperature required to decrease the resistance for a carbon resistor by 10%?