Chapter 17

Current and Resistance

Volta discovered that electricity could be created if dissimilar metals were connected by a conductive solution called an electrolyte.

This is a simple electric cell.

A battery transforms chemical energy into electrical energy.

Chemical reactions within the cell create a potential difference between the terminals by slowly dissolving them. This potential difference can be maintained even if a current is kept flowing, until one or the other terminal is completely dissolved.

Several cells connected together make a battery, although now we refer to a single cell as a battery as well.

Electric Current The current is the rate at which the

charge flows through this surface Look at the charges flowing perpendicularly

to a surface of area A

The SI unit of current is Ampere (A) 1 A = 1 C/s

QI

t

Active Figure

Electric Current, cont The direction of the current is the

direction positive charge would flow This is known as conventional current

direction In a common conductor, such as copper, the

current is due to the motion of the negatively charged electrons

It is common to refer to a moving charge as a mobile charge carrier A charge carrier can be positive or negative

By convention, current is defined as flowing from + to -. Electrons actually flow in the opposite direction, but not all currents consist of electrons.

Quick QuizConsider positive and negative charges moving horizontally through the four regions in Figure 17.2. Rank the magnitudes of the currents in these four regions from lowest to highest. (Ia is the current in Figure 17.2a, Ib the current in Figure 17.2b, etc.) (a) Id , Ia , Ic , Ib (b) Ia , Ic , Ib , Id (c) Ic , Ia , Id , Ib (d) Id , Ib , Ic , Ia (e) Ia , Ib , Ic , Id (f) none of these

Answer

(d). Negative charges moving in one direction are equivalent to positive charges moving in the opposite direction. Thus, are equivalent to the movement of 5, 3, 4, and 2 charges respectively, giving .

Current and Drift Speed

Charged particles move through a conductor of cross-sectional area A

n is the number of charge carriers per unit volume

n A Δx is the total number of charge carriers

Current and Drift Speed, cont The total charge is the number of

carriers times the charge per carrier, q ΔQ = (n A Δx) q

The drift speed, vd, is the speed at which the carriers move vd = Δx/ Δt

Rewritten: ΔQ = (n A vd Δt) q Finally, current, I = ΔQ/Δt = nqvdA

Current and Drift Speed, final If the conductor is isolated, the

electrons undergo random motion When an electric field is set up in

the conductor, it creates an electric force on the electrons and hence a current

Active Figure

Electrons in a conductor have large, random speeds just due to their temperature. When a potential difference is applied, the electrons also acquire an average drift velocity, which is generally considerably smaller than the thermal velocity.

Charge Carrier Motion in a Conductor

The zig-zag black line represents the motion of charge carrier in a conductor

The net drift speed is small

The sharp changes in direction are due to collisions

The net motion of electrons is opposite the direction of the electric field

Electrons in a Circuit The drift speed is much smaller than the

average speed between collisions When a circuit is completed, the electric

field travels with a speed close to the speed of light

Although the drift speed is on the order of 10-4 m/s the effect of the electric field is felt on the order of 108 m/s

Quick Quiz

Suppose a current-carrying wire has a cross-sectional area that gradually becomes smaller along the wire, so that the wire has the shape of a very long cone. How does the drift speed vary along the wire? (a) It slows down as the cross section becomes smaller. (b) It speeds up as the cross section becomes smaller. (c) It doesn’t change. (d) More information is needed.

Answer

(b). Under steady-state conditions, the current is the same

in all parts of the wire. Thus, the drift velocity, given by

,is inversely proportional to the cross-sectional area.

dv I nqA

Circuits

A circuit is a complete conducting path.

Meters in a Circuit – Ammeter

An ammeter is used to measure current In line with the bulb, all the charge passing

through the bulb also must pass through the meter

Meters in a Circuit – Voltmeter

A voltmeter is used to measure voltage (potential difference) Connects to the two ends of the bulb

Quick Quiz

Look at the four “circuits” shown in Figure 17.6 and select those that will light the bulb.

Answer

(c), (d). Neither circuit (a) nor circuit (b) applies a difference in potential across the bulb. Circuit (a) has both lead wires connected to the same battery terminal. Circuit (b) has a low resistance path (a “short”) between the two battery terminals as well as between the bulb terminals.

Quick Quiz

Suppose an electrical wire is replaced with one having every linear dimension doubled (i.e. the length and radius have twice their original values). Does the wire now have (a) more resistance than before, (b) less resistance, or (c) the same resistance?

Answer

(b). Consider the expression for resistance:

Doubling all linear dimensions increases the numerator of this expression by a factor of 2, but increases the denominator by a factor of 4. Thus, the net result is that the resistance will be reduced to one-half of its original value.

2RA r

l l

Example 1

A certain conductor has 7.50 × 1028 free electrons per cubic meter, a cross-sectional area of 4.00 × 10−6 m2, and carries a current of 2.50 A. Find the drift speed of the electrons in the conductor.

Example 2

In a particular television picture tube, the measured beam current is 60.0 μA. How many electrons strike the screen every second?

Example 3

An aluminum wire with a cross-sectional area of 4.0 × 10−6 m2 carries a current of 5.0 A. Find the drift speed of the electrons in the wire. The density of aluminum is 2.7 g/cm3. (Assume that one electron is supplied by each atom.)

Practice 1

A 200-km-long high-voltage transmission line 2.0 cm in diameter carries a steady current of 1 000 A. If the conductor is copper with a free charge density of 8.5 × 1028 electrons per cubic meter, how many years does it take one electron to travel the full length of the cable?

Resistance In a conductor, the voltage applied

across the ends of the conductor is proportional to the current through the conductor

The constant of proportionality is the resistance of the conductor

VR

I

Active Figure

Resistance, cont Units of resistance are ohms (Ω)

1 Ω = 1 V / A Resistance in a circuit arises due to

collisions between the electrons carrying the current with the fixed atoms inside the conductor

Georg Simon Ohm 1787 – 1854 Formulated the

concept of resistance

Discovered the proportionality between current and voltages

Ohm’s Law Experiments show that for many materials,

including most metals, the resistance remains constant over a wide range of applied voltages or currents

This statement has become known as Ohm’s Law ΔV = I R

Ohm’s Law is an empirical relationship that is valid only for certain materials Materials that obey Ohm’s Law are said to be

ohmic

Ohm’s Law, cont

An ohmic device The resistance is

constant over a wide range of voltages

The relationship between current and voltage is linear

The slope is related to the resistance

Ohm’s Law, final Non-ohmic materials

are those whose resistance changes with voltage or current

The current-voltage relationship is nonlinear

A diode is a common example of a non-ohmic device

Resistivity The resistance of an ohmic conductor is

proportional to its length, L, and inversely proportional to its cross-sectional area, A

ρ is the constant of proportionality and is called the resistivity of the material

See table 17.1

LR

A

Temperature Variation of Resistivity For most metals, resistivity

increases with increasing temperature With a higher temperature, the

metal’s constituent atoms vibrate with increasing amplitude

The electrons find it more difficult to pass through the atoms

Temperature Variation of Resistivity, cont For most metals, resistivity increases

approximately linearly with temperature over a limited temperature range

ρ is the resistivity at some temperature T ρo is the resistivity at some reference

temperature To To is usually taken to be 20° C is the temperature coefficient of resistivity

)]TT(1[ oo

Temperature Variation of Resistance Since the resistance of a conductor

with uniform cross sectional area is proportional to the resistivity, you can find the effect of temperature on resistance

)]TT(1[RR oo

Two wires, Two wires, AA and and BB, are made of , are made of

the same metal and have equal the same metal and have equal

length, but the resistance of wire length, but the resistance of wire AA

is four times the resistance of wire is four times the resistance of wire

BB. How do their diameters . How do their diameters

compare?compare?

1) ddAA = 4 = 4 ddBB

2) ddAA = 2 = 2 ddBB

3) ddAA = = ddBB

4) 4) ddAA = 1/2 = 1/2 ddBB

5) 5) ddAA = 1/4 = 1/4 ddBB

Quick Quiz

The resistance of wire A is greater because its area is lessarea is less than

wire B. Since areaarea is related to radiusradius (or diameter) squaredsquared,

the diameter of diameter of AA must be two times less than must be two times less than BB.

Two wires, Two wires, AA and and BB, are made of the , are made of the

same metalsame metal and have and have equal lengthequal length, but , but

the resistance of wire the resistance of wire AA is is four timesfour times

the resistance of wire the resistance of wire BB. How do their . How do their

diameters compare?diameters compare?

1) ddAA = 4 = 4 ddBB

2) ddAA = 2 = 2 ddBB

3) ddAA = = ddBB

4) 4) ddAA = 1/2 = 1/2 ddBB

5) 5) ddAA = 1/4 = 1/4 ddBB

Answer

A wire of resistance A wire of resistance RR is is

stretched uniformly (keeping stretched uniformly (keeping

its volume constant) until it is its volume constant) until it is

twice its original length. twice its original length.

What happens to the What happens to the

resistance?resistance?

1) it decreasesit decreases by a factor 4by a factor 4

2) it decreasesit decreases by a factor 2by a factor 2

3) it stays the sameit stays the same

4) it increases4) it increases by a factor 2by a factor 2

5) it increases5) it increases by a factor 4by a factor 4

Quick Quiz

Keeping the volume (= area x length) constant means that if the

length is doubleddoubled, the area is halvedhalved.

Since R=L/A , this increases the resistance by fourfour.

A wire of resistance A wire of resistance RR is is

stretched uniformly (keeping stretched uniformly (keeping

its volume constant) until it is its volume constant) until it is

twice its original length. What twice its original length. What

happens to the resistance?happens to the resistance?

1) it decreasesit decreases by a factor 4by a factor 4

2) it decreasesit decreases by a factor 2by a factor 2

3) it stays the sameit stays the same

4) it increases4) it increases by a factor 2by a factor 2

5) it increases5) it increases by a factor 4by a factor 4

Answer

Example 4

A lightbulb has a resistance of 240 Ω when operating at a voltage of 120 V. What is the current in the bulb?

Example 5

Eighteen-gauge wire has a diameter of 1.024 mm. Calculate the resistance of 15 m of 18-gauge copper wire at 20°C.

Example 6

A rectangular block of copper has sides of length 10 cm, 20 cm, and 40 cm. If the block is connected to a 6.0-V source across two of its opposite faces, what are (a) the maximum current and (b) the minimum current that the block can carry?

Practice 2

Suppose that you wish to fabricate a uniform wire out of 1.00 g of copper. If the wire is to have a resistance R = 0.500 Ω, and if all of the copper is to be used, what will be (a) the length and (b) the diameter of the wire?

Example 7

A wire 3.00 m long and 0.450 mm2 in cross-sectional area has a resistance of 41.0 Ω at 20°C. If its resistance increases to 41.4 Ω at 29.0°C, what is the temperature coefficient of resistivity?

Example 8

A 100-cm-long copper wire of radius 0.50 cm has a potential difference across it sufficient to produce a current of 3.0 A at 20°C. (a) What is the potential difference? (b) If the temperature of the wire is increased to 200°C, what potential difference is now required to produce a current of 3.0 A?

Practice 3

The copper wire used in a house has a cross-sectional area of 3.00 mm2. If 10.0 m of this wire is used to wire a circuit in the house at 20.0°C, find the resistance of the wire at temperatures of (a) 30.0°C and (b) 10.0°C.

Superconductors A class of materials

and compounds whose resistances fall to virtually zero below a certain temperature, TC

TC is called the critical temperature

The graph is the same as a normal metal above TC, but suddenly drops to zero at TC

Superconductors, cont The value of TC is sensitive to

Chemical composition Pressure Crystalline structure

Once a current is set up in a superconductor, it persists without any applied voltage Since R = 0

Superconductor Timeline 1911

Superconductivity discovered by H. Kamerlingh Onnes

1986 High temperature superconductivity discovered

by Bednorz and Müller Superconductivity near 30 K

1987 Superconductivity at 96 K and 105 K

Current More materials and more applications

Superconductor, final Good conductors

do not necessarily exhibit superconductivity

One application is superconducting magnets

Electrical Energy and Power In a circuit, as a charge moves through

the battery, the electrical potential energy of the system is increased by ΔQΔV The chemical potential energy of the

battery decreases by the same amount As the charge moves through a resistor,

it loses this potential energy during collisions with atoms in the resistor The temperature of the resistor will increase

Energy Transfer in the Circuit Consider the

circuit shown Imagine a

quantity of positive charge, Q, moving around the circuit from point A back to point A

Energy Transfer in the Circuit, cont Point A is the reference point

It is grounded and its potential is taken to be zero

As the charge moves through the battery from A to B, the potential energy of the system increases by QV The chemical energy of the battery

decreases by the same amount

Energy Transfer in the Circuit, final As the charge moves through the

resistor, from C to D, it loses energy in collisions with the atoms of the resistor

The energy is transferred to internal energy

When the charge returns to A, the net result is that some chemical energy of the battery has been delivered to the resistor and caused its temperature to rise

Electrical Energy and Power, cont The rate at which the energy is lost

is the power

From Ohm’s Law, alternate forms of power are

QV I V

t

22 V

I RR

Electrical Energy and Power, final The SI unit of power is Watt (W)

I must be in Amperes, R in ohms and V in Volts

The unit of energy used by electric companies is the kilowatt-hour This is defined in terms of the unit of

power and the amount of time it is supplied

1 kWh = 3.60 x 106 J

Active Figure

When you rotate the knob of a When you rotate the knob of a

light dimmer, what is being light dimmer, what is being

changed in the electric circuit?changed in the electric circuit?

1) the power

2) the current

3) the voltage

4) both (1) and (2)

5) both (2) and (3)

Quick Quiz

The voltage is provided at 120 V from the

outside. The light dimmer increases the increases the

resistanceresistance and therefore decreases the decreases the

currentcurrent that flows through the lightbulb.

When you rotate the knob of a When you rotate the knob of a

light dimmer, what is being light dimmer, what is being

changed in the electric circuit?changed in the electric circuit?

1) the power

2) the current

3) the voltage

4) both (1) and (2)

5) both (2) and (3)

Answer

Two lightbulbs operate at 120 V, Two lightbulbs operate at 120 V,

but one has a power rating of but one has a power rating of 25 W25 W

while the other has a power rating while the other has a power rating

of of 100 W100 W. Which one has the . Which one has the

greater resistance? greater resistance?

1) the 25 W bulb

2) the 100 W bulb

3) both have the same

4) this has nothing to do with resistance

Quick Quiz

Since P = VP = V22 / R / R the bulb with the lower lower

power ratingpower rating has to have the higher higher

resistanceresistance.

Two lightbulbs operate at 120 V, but Two lightbulbs operate at 120 V, but

one has a power rating of one has a power rating of 25 W25 W while while

the other has a power rating of the other has a power rating of 100 W100 W. .

Which one has the greater resistance? Which one has the greater resistance?

1) the 25 W bulb

2) the 100 W bulb

3) both have the same

4) this has nothing to do with resistance

Answer

Two space heaters in your Two space heaters in your

living room are operated at living room are operated at

120 V. Heater 1 has 120 V. Heater 1 has twicetwice

the resistance of heater 2. the resistance of heater 2.

Which one will give off more Which one will give off more

heat?heat?

1) heater 1

2) heater 2

3) both equally

Quick Quiz

Using P = VP = V22 / R, / R, the heater with the smaller resistancesmaller resistance

will have the larger powerlarger power output. Thus, heater 2 will

give off more heat.

Two space heaters in your Two space heaters in your

living room are operated at 120 living room are operated at 120

V. Heater 1 has V. Heater 1 has twicetwice the the

resistance of heater 2. Which resistance of heater 2. Which

one will give off more heat?one will give off more heat?

1) heater 1

2) heater 2

3) both equally

Answer

A voltage ΔV is applied across the ends of a nichrome heater wire having a cross-sectional area A and length L. The same voltage is applied across the ends of a second heater wire having a cross-sectional area A and length 2L. Which wire gets hotter? (a) the shorter wire, (b) the longer wire, or (c) more information is needed.

Quick Quiz

Answer

(a). The resistance of the shorter wire is half that of the longer wire. The power dissipated, , (and hence the rate of heating) will be greater for the shorter wire. Consideration of the expression might initially lead one to think that the reverse would be true. However, one must realize that the currents will not be the same in the two wires.

2V R P

2I RP

Quick Quiz

For the two resistors shown in Figure 17.12, rank the currents at points a through f from largest to smallest.(a) Ia = Ib > Ie = If > Ic = Id(b) Ia = Ib > Ic = Id > Ie = If(c) Ie = If > Ic = Id > Ia = Ib

Answer

(b). . Charges constituting the current

leave the positive terminal of the battery and then split to flow through the two bulbs; thus, . Because the potential difference is the same across the two bulbs and because the power delivered to a device is , the 60–W bulb with the higher power rating must carry the greater current, meaning that . Because charge does not accumulate in the bulbs, all the charge flowing into a bulb from the left has to flow out on the right; consequently and . The two currents leaving the bulbs recombine to form the current back into the battery, .

a b c d e fI I I I I I

aI

a c eI I I

I V P

c eI I

c dI I e fI I

f d bI I I

Active Figure

Example 9

A high-voltage transmission line with a resistance of 0.31 Ω/km carries a current of 1 000 A. The line is at a potential of 700 kV at the power station and carries the current to a city located 160 km from the station. (a) What is the power loss due to resistance in the line? (b) What fraction of the transmitted power does this loss represent?

Example 10

What is the required resistance of an immersion heater that will increase the temperature of 1.50 kg of water from 10.0°C to 50.0°C in 10.0 min while operating at 120 V?

Example 11

How much does it cost to watch a complete 21-hour-long World Series on a 180-W television set? Assume that electricity costs $0.070/kWh.

Example 12

A house is heated by a 24.0-kW electric furnace that uses resistance heating. The rate for electrical energy is $0.080/kWh. If the heating bill for January is $200, how long must the furnace have been running on an average January day?

Practice 4

It has been estimated that there are 270 million plug-in electric clocks in the United States, approximately one clock for each person. The clocks convert energy at the average rate of 2.50 W. To supply this energy, how many metric tons of coal are burned per hour in coal-fired electric generating plants that are, on average, 25.0% efficient? The heat of combustion for coal is 33.0 MJ/kg.