Chapter 5 Review

Key Terms

ampere (amp)
SI unit for current;

circuit
complete path that an electrical current travels along

conventional current
current that flows through a circuit from the positive terminal of a battery through the circuit to the negative terminal of the battery

critical temperature
temperature at which a material reaches superconductivity

current density
flow of charge through a cross-sectional area divided by the area

diode
nonohmic circuit device that allows current flow in only one direction

drift velocity
velocity of a charge as it moves nearly randomly through a conductor, experiencing multiple collisions, averaged over a length of a conductor, whose magnitude is the length of conductor traveled divided by the time it takes for the charges to travel the length

electrical conductivity
measure of a material’s ability to conduct or transmit electricity

electrical current
rate at which charge flows,

electrical power
time rate of change of energy in an electric circuit

Josephson junction
junction of two pieces of superconducting material separated by a thin layer of insulating material, which can carry a supercurrent

Meissner effect
phenomenon that occurs in a superconducting material where all magnetic fields are expelled

nonohmic
type of a material for which Ohm’s law is not valid

ohm
(

) unit of electrical resistance,

ohmic
type of a material for which Ohm’s law is valid, that is, the voltage drop across the device is equal to the current times the resistance

Ohm’s law
empirical relation stating that the current

is proportional to the potential difference

; it is often written as

where

is the resistance

resistance
electric property that impedes current; for ohmic materials, it is the ratio of voltage to current,

resistivity
intrinsic property of a material, independent of its shape or size, directly proportional to the resistance, denoted by

schematic
graphical representation of a circuit using standardized symbols for components and solid lines for the wire connecting the components

SQUID
(Superconducting Quantum Interference Device) device that is a very sensitive magnetometer, used to measure extremely subtle magnetic fields

superconductivity
phenomenon that occurs in some materials where the resistance goes to exactly zero and all magnetic fields are expelled, which occurs dramatically at some low critical temperature (

)

Key Equations

Average electrical current	$I_{\mathrm{ave}}=\frac{\Delta Q}{\Delta t}$
Definition of an ampere	$1~\mathrm{A}=1~\mathrm{C/s}$
Electrical current	$I=\frac{dQ}{dt}$
Drift velocity	$v_d=\frac{I}{nqA}$
Current density	$I=\iint_{\mathrm{area}}\vec{\mathbf{J}}\cdot d\vec{\mathbf{A}}$
Resistivity	$\rho=\frac{E}{J}$
Common expression of Ohm’s law	$V=IR$
Resistivity as a function of temperature	$\rho=\rho_0[1+\alpha(T-T_0)]$
Definition of resistance	$R\equiv\frac{V}{I}$
Resistance of a cylinder of material	$R=\rho\frac{L}{A}$
Temperature dependence of resistance	$R=R_0(1+\alpha\Delta T)$
Electric power	$P=IV$
Power dissipated by a resistor	$P=I^2R=\frac{V^2}{R}$

Summary

5.1 Electrical Current

The average electrical current $I_{\mathrm{ave}}$ is the rate at which charge flows, given by $I_{\mathrm{ave}}=\frac{\Delta Q}{\Delta t}$ where $\Delta Q$ is the amount of charge passing through an area in time $\Delta t.$
The instantaneous electrical current, or simply the current $I,$ is the rate at which charge flows. Taking the limit as the change in time approaches zero, we have $I=\frac{dQ}{dt},$ where $\frac{dQ}{dt}$ is the time derivative of the charge.
The direction of conventional current is taken as the direction in which positive charge moves. In a simple direct-current (DC) circuit, this will be from the positive terminal of the battery to the negative terminal.
The SI unit for current is the ampere, or simply the amp ( $\mathrm{A}$ ), where $1~\mathrm{A}=1~\mathrm{C/s}.$
Current consists of the flow of free charges, such as electrons, protons, and ions.

5.2 Model of Conduction in Metals

The current through a conductor depends mainly on the motion of free electrons.
When an electrical field is applied to a conductor, the free electrons in a conductor do not move through a conductor at a constant speed and direction; instead, the motion is almost random due to collisions with atoms and other free electrons.
Even though the electrons move in a nearly random fashion, when an electrical field is applied to the conductor, the overall velocity of the electrons can be defined in terms of a drift velocity.
The current density is a vector quantity defined as the current through an infinitesimal area divided by the area.
The current can be found from the current density, $I=\iint_{\mathrm{area}}\vec{\mathbf{J}}\cdot d\vec{\mathbf{A}}.$
An incandescent light bulb is a filament of wire enclosed in a glass bulb that is partially evacuated. Current runs through the filament, where the electrical energy is converted to light and heat.

5.3 Resistivity and Resistance

Resistance has units of ohms ( $\Omega$ ), related to volts and amperes by $1~\mathrm{\Omega}=1~\mathrm{V/A}.$
The resistance $R$ of a cylinder of length $L$ and cross-sectional area $A$ is $R=\frac{\rho L}{A},$ where $\rho$ is the resistivity of the material.
Values of $\rho$ in Table 5.3.1 show that materials fall into three groups—conductors, semiconductors, and insulators.
Temperature affects resistivity; for relatively small temperature changes $\Delta T,$ resistivity is $\rho=\rho_0(1+\alpha\Delta T),$ where $\rho_0$ is the original resistivity and $\alpha$ is the temperature coefficient of resistivity.
The resistance $R$ of an object also varies with temperature: $R=R_0(1+\alpha\Delta T)$ where $R_0$ is the original resistance, and $R$ is the resistance after the temperature change.

5.4 Ohm’s Law

Ohm’s law is an empirical relationship for current, voltage, and resistance for some common types of circuit elements, including resistors. It does not apply to other devices, such as diodes.
One statement of Ohm’s law gives the relationship among current $I,$ voltage $V,$ and resistance $R$ in a simple circuit as $V=IR.$
Another statement of Ohm’s law, on a microscopic level, is $J=\sigma E.$

5.5 Electrical Energy and Power

Electric power is the rate at which electric energy is supplied to a circuit or consumed by a load.
Power dissipated by a resistor depends on the square of the current through the resistor and is equal to $P=I^2R=\frac{V^2}{R}.$
The SI unit for electric power is the watt and the SI unit for electric energy is the joule. Another common unit for electric energy, used by power companies, is the kilowatt-hour ( $\mathrm{kW}\cdot\mathrm{h}$ ).
The total energy used over a time interval can be found by $E=\int Pdt.$

5.6 Superconductors

Superconductivity is a phenomenon that occurs in some materials when cooled to very low critical temperatures, resulting in a resistance of exactly zero and the expulsion of all magnetic fields.
Materials that are normally good conductors (such as copper, gold, and silver) do not experience superconductivity.
Superconductivity was first observed in mercury by Heike Kamerlingh Onnes in 1911. In 1986, Dr. Ching Wu Chu of Houston University fabricated a brittle, ceramic compound with a critical temperature close to the temperature of liquid nitrogen.
Superconductivity can be used in the manufacture of superconducting magnets for use in MRIs and high-speed, levitated trains.

Answers to Check Your Understanding

5.1 The time for

of charge to flow would be

slightly less than an hour. This is quite different from the

for the truck battery. The calculator takes a very small amount of energy to operate, unlike the truck’s starter motor. There are several reasons that vehicles use batteries and not solar cells. Aside from the obvious fact that a light source to run the solar cells for a car or truck is not always available, the large amount of current needed to start the engine cannot easily be supplied by present-day solar cells. Solar cells can possibly be used to charge the batteries. Charging the battery requires a small amount of energy when compared to the energy required to run the engine and the other accessories such as the heater and air conditioner. Present day solar-powered cars are powered by solar panels, which may power an electric motor, instead of an internal combustion engine.

5.2 The total current needed by all the appliances in the living room (a few lamps, a television, and your laptop) draw less current and require less power than the refrigerator.

5.3 The diameter of the

-gauge wire is smaller than the diameter of the

-gauge wire. Since the drift velocity is inversely proportional to the cross-sectional area, the drift velocity in the

-gauge wire is larger than the drift velocity in the

-gauge wire carrying the same current. The number of electrons per cubic meter will remain constant.

5.4 The current density in a conducting wire increases due to an increase in current. The drift velocity is inversely proportional to the current

so the drift velocity would decrease.

5.5 Silver, gold, and aluminum are all used for making wires. All four materials have a high conductivity, silver having the highest. All four can easily be drawn into wires and have a high tensile strength, though not as high as copper. The obvious disadvantage of gold and silver is the cost, but silver and gold wires are used for special applications, such as speaker wires. Gold does not oxidize, making better connections between components. Aluminum wires do have their drawbacks. Aluminum has a higher resistivity than copper, so a larger diameter is needed to match the resistance per length of copper wires, but aluminum is cheaper than copper, so this is not a major drawback. Aluminum wires do not have as high of a ductility and tensile strength as copper, but the ductility and tensile strength is within acceptable levels. There are a few concerns that must be addressed in using aluminum and care must be used when making connections. Aluminum has a higher rate of thermal expansion than copper, which can lead to loose connections and a possible fire hazard. The oxidation of aluminum does not conduct and can cause problems. Special techniques must be used when using aluminum wires and components, such as electrical outlets, must be designed to accept aluminum wires.

5.6 The foil pattern stretches as the backing stretches, and the foil tracks become longer and thinner. Since the resistance is calculated as

the resistance increases as the foil tracks are stretched. When the temperature changes, so does the resistivity of the foil tracks, changing the resistance. One way to combat this is to use two strain gauges, one used as a reference and the other used to measure the strain. The two strain gauges are kept at a constant temperature

5.7 The longer the length, the smaller the resistance. The greater the resistivity, the higher the resistance. The larger the difference between the outer radius and the inner radius, that is, the greater the ratio between the two, the greater the resistance. If you are attempting to maximize the resistance, the choice of the values for these variables will depend on the application. For example, if the cable must be flexible, the choice of materials may be limited.

5.8 Yes, Ohm’s law is still valid. At every point in time the current is equal to

so the current is also a function of time,

5.9 Even though electric motors are highly efficient

of the power consumed is wasted, not being used for doing useful work. Most of the

of the power lost is transferred into heat dissipated by the copper wires used to make the coils of the motor. This heat adds to the heat of the environment and adds to the demand on power plants providing the power. The demand on the power plant can lead to increased greenhouse gases, particularly if the power plant uses coal or gas as fuel.

5.10 No, the efficiency is a very important consideration of the light bulbs, but there are many other considerations. As mentioned above, the cost of the bulbs and the life span of the bulbs are important considerations. For example, CFL bulbs contain mercury, a neurotoxin, and must be disposed of as hazardous waste. When replacing incandescent bulbs that are being controlled by a dimmer switch with LED, the dimmer switch may need to be replaced. The dimmer switches for LED lights are comparably priced to the incandescent light switches, but this is an initial cost which should be considered. The spectrum of light should also be considered, but there is a broad range of color temperatures available, so you should be able to find one that fits your needs. None of these considerations mentioned are meant to discourage the use of LED or CFL light bulbs, but they are considerations.