Current and

Resistance

The Starting Point: Elements, Atoms and Charge

Electrons and protons have, in addition totheir mass, a quantity called charge

Charge (unlike mass) can be either positive(protons) or negative (electrons)

Like charges repel, unlike charges attract

Free ElectronsAn electron that is not bound to any particular atom

Ions External force can cause an electron to

leave its orbit -atom is referred to as a positive ion

External force can cause an atom to gain an electron -atom is referred to as a negative ion

Electric Current Whenever electric charges, an electric

current is said to exist 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

Electrical Current

Insert Figure 1.10

Electron Flow Versus Conventional Current

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

Current and Drift Speed 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

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

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

nAΔx is the total number of charge carriers

Current and Drift Speed 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

Quiz 1 Consider positive and negative charges moving

horizontally through the four regions in the following Figure. Rank the magnitudes of the currents in these four regions from lowest to highest.

(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 Quiz 1

(d). Negative charges moving in one direction are equivalent to positive charges moving in the opposite direction. Thus, Ia, Ib, Ic and Id

are equivalent to the movement of 5, 3, 4, and 2 charges respectively, giving Id < Ib < Ic< Ia

A copper wire of cross-sectional area 3.00x10-6 m2 carries a current of 10 A. Assuming that each copper atom contributes one free electron to the metal, find the drift speed of the electron in this wire. The density of copper is 8.95 g/cm3.

22 3 19 6 2

6

10.0 /

8.48 10 1.6 10 3.00 10

2.46 10 /

d

I C sv

nqA electrons m C m

m s

Example

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

Quiz 2 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 Quiz 2

(b). Under steady-state conditions, the current is the same in all parts of the wire. Thus, the drift velocity, given by vd = I/nqA, is inversely proportional to the cross-sectional area

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

Voltage

The electrical potential difference is what causes current to flow.

The basic unit of voltage is the volts . ~

Current flow-Amperes (Amps)

Ampere: unit of measurement for electrical current.

One volt across aresistance of one ohm will produce a flow of one amp. ~

Exercise The capacity of a car battery is usually

specified in ampere-hours. A battery rated at, say, 100 A-h should be able to supply 100 A for 1 h, 50 A for 2 h, 25 A for 4 h, 1 A for 100 h, or any other combination yielding a product of 100 A-h. a. How many coulombs of charge should we be

able to draw from a fully charged 100 A-h battery?

b. How many electrons does your answer to part a require?

Answer

100 1 100 (1 ) 3600 360000C s

A hr hr Cs hr

322

19

360 10224.7 10

1.602 10

Assumptions:Battery fully charged.

a)

b) charge on electron: -1.602 ´ 10-19 Cno. of electrons =

Ohm’s Law German Physicist – George Simon Ohm

Found that current is inversely proportional to resistance for a given voltage

Known as Ohm’s law The Relationship Between Current and Voltage The Relationship Between Current and Resistance

Basic Circuit Calculations

Using Ohm’s Law to Calculate Current

where R = the circuit resistanceV = the applied voltage

R

VI

Basic Circuit Calculations Using Ohm’s Law to Calculate

Voltage

where I = the circuit currentR = the circuit resistance

IRV

Basic Circuit Calculations

Using Ohm’s Law to Calculate Resistance

whereV = the circuit voltage I = the circuit current

I

VR

Resistance Element

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 LawOhm’s LawConductivity and resistivity

VL

AI

EA

IJ

1

A

LIV

IRV

AR

is resistivity

is current density

Example

A nichrome wire of radius 0.321 mm has resistivity of 1.5 x 10-6 Ωm. If a potential difference of 10.0 V is maintained across a 1 m length of the nichrome wire, what is the current?

Cross section: 22 3 7 20.321 10 3.24 10A r m m 6

7 2

1.5 104.6

3.24 10m

R m

l A m

Resistance/unit length:

10.02.2

4.6

V VI A

R

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

Ohmic devices vs. non-Ohmic

Quiz 4 In Figure, does

the resistance of the diode (a) increase or (b) decrease as the positive voltage ΔV increases?

Answer Quiz 4

(b). The slope of the line tangent to the curve at a point is the reciprocal of the resistance at that point. Note that as increases, the slope (and hence ) increases. Thus, the resistance decreases.

Temperature Dependence of Temperature Dependence of ResistanceResistance

Define temp coefficient of resistivityDefine temp coefficient of resistivity

dTd1

If If is small and constant is small and constant

oo TT1

Units of Units of is ( is (ooC)C)-1-1

Platinum Resistance ThermometerA resistance thermometer, which measures temperature by measuring the change in the resistance of a conductor, is made of platinum and has a resistance of 50 Ω at 20oC. When the device is immersed in a vessel containing melting indium, its resistance increases to 76.8 Ω. Find the melting point of Indium.

CR

RRTT o

o

oo 137

CT o157

Example

αα=3.92x10=3.92x10-3-3((ooC)C)-1-1

Since the To= 20oc then

Microscopic view of Ohm’s Law

2ne

mResistivity is a constant

Free electron Theory

Why do old light bulbs give less light than when new?

The filament of a light bulb, made of tungsten, is kept at high temperature when the light bulb is on.

It tends to evaporate, i.e. to become thinner, thus decreasing in radius, and cross sectional area.

Its resistance increases with time. The current going though the filament then decreases with time – and so does its luminosity.

Tungsten atoms evaporate off the filament and end up on the inner surface of the bulb.

Over time, the glass becomes less transparent and therefore less luminous.

Low temperature behavior of Low temperature behavior of resistanceresistance

SuperconductivitySuperconductivitySemiconductorSemiconductor

Exercise 1 If a current of 80.0 mA exists in a

metal wire, how many electrons flow past a given cross section of the wire in 10.0 min? Sketch the direction of the current and the direction of the electrons’ motion.

Exercise 2 If 3.25 × 10−3 kg of gold is

deposited on the negative electrode of an electrolytic cell in a period of 2.78 h, what is the current in the cell during that period? Assume that the gold ions carry one elementary unit of positive charge.

Exercise 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.)

Exercise 4 If the current carried by a

conductor is doubled, what happens to (a) the charge carrier density? (b) the electron drift velocity?