Chapter 9 Current and Resistancecremaldi/PHYS212/ch26(9).pdfThis is a statement of Ohm's law (the...

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1 Chapter 9 Current and Resistance In this chapter we will introduce the following new concepts: -Electric current ( symbol i ) -Electric current density vector (symbol ) -Drift speed (symbol υ d ) -Resistance (symbol R ) and resistivity ( ρ ) and conductivity( σ ) of a conductor -Ohmic and non-Ohmic conductors We will also cover the following topics: -Ohms law -Power in electric circuits J !

Transcript of Chapter 9 Current and Resistancecremaldi/PHYS212/ch26(9).pdfThis is a statement of Ohm's law (the...

Page 1: Chapter 9 Current and Resistancecremaldi/PHYS212/ch26(9).pdfThis is a statement of Ohm's law (the resistance of the conductor does not depend on voltage and thus ) This is because

1

Chapter 9 Current and Resistance In this chapter we will introduce the following new concepts:

-Electric current ( symbol i ) -Electric current density vector (symbol ) -Drift speed (symbol υd ) -Resistance (symbol R ) and resistivity ( ρ ) and conductivity( σ ) of a conductor -Ohmic and non-Ohmic conductors

We will also cover the following topics:

-Ohm’s law -Power in electric circuits

J!

Page 2: Chapter 9 Current and Resistancecremaldi/PHYS212/ch26(9).pdfThis is a statement of Ohm's law (the resistance of the conductor does not depend on voltage and thus ) This is because

Electric currentConsider the conductor shown in fig.a. All the points inside the conductor and on its surface are at the same potential. The free electrons inside the conductor move at random directions and thus there is not net charge transport. We now make a break in the conductorand insert a battery as shown in fig.b.Points A and B are now at potentials VA and VB , respectively.

VA −VB =V the voltage of the battery)

The situation is not static any more but charges move inside the conductor so thatthere is a net charge flow in a particulardirection. This net flow of electric charge we define as "electric current"

A B

2

dqidt

=

i = 0

Current = rate at which charge flows Current SI Unit: C/s known as the "Ampere"

electron drift

Page 3: Chapter 9 Current and Resistancecremaldi/PHYS212/ch26(9).pdfThis is a statement of Ohm's law (the resistance of the conductor does not depend on voltage and thus ) This is because

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Driftspeedυd acloserlook

12mυ

2

K!

= qeVU!

free electron transport

12mυ

2 = qeV −Q transport with collision losses

υd~2(qeV −Q)

me

Q = energy of resistive losses converted to heat

V

+ !υ

↘↘↘Q-loss

Current direction :1. The electric current is defined to flow inthe direction of positive charges from the + battery terminal.2. True electrons flow from the - battery terminalto the positive terminal. Electron flow can be thought of as a +holemoving from the + terminal.

e-

V

K +U=K’+ U’ qV=1/2mυ2

+ -

+

Page 4: Chapter 9 Current and Resistancecremaldi/PHYS212/ch26(9).pdfThis is a statement of Ohm's law (the resistance of the conductor does not depend on voltage and thus ) This is because

Current densityCurrent density is a vector that is defined as follows:

Its magnitude J = iA

(A/m2 )

The direction of !J is the same as that of the current

The current through a conductor of cross sectionalarea A is given by the equation: i = JA if the current density is constant.

If !J is not constant then: i =

!J ⋅d!A∫

4

The current density is:

J = iA= dq / dt

A= (

# chargesV

)q ⋅ dxdt

= neυd →!J = ne

!υd

n ≡ρN A

Α= density × Avogadro 's#

atomic number= (

#V

) number density of charge carriers

By measuring J for a wire sample of cross sectional area A the material drift speed υd can be determined.

i

+ q

conductor

!υ A

J!dx

Page 5: Chapter 9 Current and Resistancecremaldi/PHYS212/ch26(9).pdfThis is a statement of Ohm's law (the resistance of the conductor does not depend on voltage and thus ) This is because

5

+ -

i

V

Resistance If we apply a voltage V across a conductora current iwill flow. We define the resistance as the ratio:

R = Vi

.

R Circuit Symbol for R

SI Unit for R : VA

= the Ohm (symbol Ω)

A conductor across which we apply a voltage V = 1 Volt and results in a current i = 1 Ampere is defined as having resistance of 1 Ω

Conductivity, Resistivity, and Resistance The ratio of current density to electric field is referred to as the conductivityof the material σ = J / E . We also define as resistivity ρ of the conductor

as the inverse ratio ρ = E / J thus, σ = 1ρ

. (ρ andσ material dependent only)

Simplifying : ρ = EJ

and get: ρ = V / L i / A

= Vi

AL= R

AL→ R = ρ L

A

L

A

Page 6: Chapter 9 Current and Resistancecremaldi/PHYS212/ch26(9).pdfThis is a statement of Ohm's law (the resistance of the conductor does not depend on voltage and thus ) This is because

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In the figure we plot the resistivity ofcopper as function of temperature . The dependence of on is almost linear. Similar dependence is observed

TT

ρ

ρ

Variation of resistivity with temperature

in many conductors.

The following empirical equation is used for many practical applications:

ρ = ρo[1+α T −To( )] The constant α is known as the

"temperature coefficient of resistivity". The constant To is a reference temperature

usually taken to be room temperature (To = 293 K ) and ρo is the resistivity

at To. For copper ρo = 1.69×10−8 Ω⋅m

Note : temperature enters the equation above as a difference T −To( ).Thus either the Celsius or the Kelvin temperature scale can be used.

ρ = ρo[1+α T −To( )]

Page 7: Chapter 9 Current and Resistancecremaldi/PHYS212/ch26(9).pdfThis is a statement of Ohm's law (the resistance of the conductor does not depend on voltage and thus ) This is because

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Ohm's Law : A resistor was defined as a conductor whose resistance does not changewith the voltage V applied across it. In fig.b we plot the current i through a resistor as function of V . The plot (known as "i -V curve" ) is a straight line that passes through the origin. Such a conductor is said to be "Ohmic" and it obeys Ohm's law that states:The current i through a conductor is proportional to the voltage V applied across it.

Iresistor =V / R (Ohmic) Idiode = I0(e+eV /kT −1) (non−Ohmic)Not all conductors obey Ohm's law (these are known as "non - Ohmic" ) An example is given in fig.c where we plot i versus V for a semiconductor diode.The ratio V / i (and thus the resistance R ) is not constant. As a matter of fact the diode does not conduct for negative voltage values.

Page 8: Chapter 9 Current and Resistancecremaldi/PHYS212/ch26(9).pdfThis is a statement of Ohm's law (the resistance of the conductor does not depend on voltage and thus ) This is because

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Lamp filament non-ohmic The light bulb is a non-ohmic device. As the voltage is increased The current and brightness increase proportionally in the “cold resistance” region as predicted I = V/R. As the filament heats its resistance increases and the current I begins to droop I < V/R.

Page 9: Chapter 9 Current and Resistancecremaldi/PHYS212/ch26(9).pdfThis is a statement of Ohm's law (the resistance of the conductor does not depend on voltage and thus ) This is because

Consider circuit shown in the figure. A battery of voltage is connected across the terminals a and b of a device. This can be a resistor, a motor, etc. The battery main

VPower in electric circuits :

tains apotential difference between the terminals a and b and thus a current flows in the circuit as shown in thefigure. During the time interval a charge movesbetween the terminals. W

Vi

dt dq idt=e note that > . a bV V

The potential energy of the charge decreases by an amount dU =VdqUsing energy conservation we conclude that the lost energy has been transfered by the battery to the device nad has been converted into some other form of energy.The rate at which energy is transfered to the device is known as "power" and it is

equal to: P = dUdt

= ddt

qV =V dqdt

= iV P= iV

SI unit for P : V ⋅A It is known as the "Watt" (symbol W).

P iV=

9

Page 10: Chapter 9 Current and Resistancecremaldi/PHYS212/ch26(9).pdfThis is a statement of Ohm's law (the resistance of the conductor does not depend on voltage and thus ) This is because

Energy is not conserved in the circuit due to resistive losses. The dissapatedpower P is lost as heat in the resistors.

For Ohmic devices(e.g. resistors) i = VR

and if we combine the equation P = iV

we get the following two equivalent expressions for the rate at which heat is

dissipated on R. P = i2R and P = V 2

R

V 2 P i R=

2

VPR

=

10

P = iV (Generalexpression forall Resistance types)

(only V=IRtypes )

Page 11: Chapter 9 Current and Resistancecremaldi/PHYS212/ch26(9).pdfThis is a statement of Ohm's law (the resistance of the conductor does not depend on voltage and thus ) This is because

11

STOP HERE

Page 12: Chapter 9 Current and Resistancecremaldi/PHYS212/ch26(9).pdfThis is a statement of Ohm's law (the resistance of the conductor does not depend on voltage and thus ) This is because

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When a current flows through a conductorthe electric field causes the charges to movewith a constant drift speed . This drift speedis superimposed on the random motion of the charges.

dv

Drift speed

Consider the conductor of cross sectional area shown in the figure. We assumethat the current in the conductor consists of positive charges. The total charge

within a length is given by:

A

q L q nA= ( ) . This charge moves through area

in a time . The current /

The current density

In vector form:

dd d

dd

d

L e AL q nALet i nAv ev t L v

nAv eiJ nv eA A

J nev

= = = =

= = =

=! !

dJ nev=! !

dJ nv e=

Page 13: Chapter 9 Current and Resistancecremaldi/PHYS212/ch26(9).pdfThis is a statement of Ohm's law (the resistance of the conductor does not depend on voltage and thus ) This is because

F!

dv!

Consider the motion of one of the free electrons.We assume that the average time between collisions with the copper atoms is equal to . The electic fieldexerts a force on the electron, resultF eE

τ= ing in an

acceleration . The drift speed is given

by the equation: ( ) d

F eEam m

eEv amττ

= =

= = eqs.1

2

2

We can also get from the equation: ( )

If we compare equations 1 and 2 we get:

If we compare the last equation with: we conclude h

t at:

d d d

d

Jv J nev vne

J eE mv

mne

E Jne m ne

E J

ττ

ρτ

ρ

= → =

⎛ ⎞= = → = ⎜ ⎟

=

=

⎝ ⎠

eqs.2

This is a statement of Ohm's law (the resistance of the

conductor does not depend on voltage and thus ) This is because , , and are constants. The time can also be considered be

ind

epende

Em n e τ

nt of since the drift spees is so much smaller than .d effE v v 13

Page 14: Chapter 9 Current and Resistancecremaldi/PHYS212/ch26(9).pdfThis is a statement of Ohm's law (the resistance of the conductor does not depend on voltage and thus ) This is because

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In order to understand why some materials such asmetals obey Ohm's law we must look into the detailsof the conduction process at the atomic level. A schematic of an O

A Microspopic view of Ohm's law :

6

hmic conductor such as copper isshown in the figure. We assume that there are free electrons that move around in random directionswith an effective speed = 1.6 10 m/s. The free

electrons suffer effv ×

collisions with the stationary copper atoms.

A schematic of a free electron path is shown in the figure in the figure using the dashedgray line. The electron starts at point A and ends at point B. We now assume that an

electric field is applE!

ied. The new electron path is indicated by the dashed green lineUnder the action of the electric force the electron acquires a small drift speed . The electron drifts to the right and ends at po

dvint B . ′

F!

dv!

Page 15: Chapter 9 Current and Resistancecremaldi/PHYS212/ch26(9).pdfThis is a statement of Ohm's law (the resistance of the conductor does not depend on voltage and thus ) This is because

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E!

+ -

i

V

E!

Unlike the electrostatic case, the electric field in theconductor of the figure is not zero. We define as

resistivity of the conductor the ratio

In vector form: EJ

J

Eρ ρ

ρ=

=

Resi

SI u

stivit

nit r

y

fo

! !

2

The conductivity is defined as:

Using the previou

V/m V m mA/m

s equation takes the for

1

m

A

: J E

σ

ρ

σρ

σ

=

= =Ω⋅

=

ñ :

! !

Consider the conductor shown in the figure above. The electric field inside the

conductor E = VL

. The current density J = iA

We substitute E and J into

equation ρ = EJ

and get: ρ = V / L i / A

= Vi

AL= R A

L→ R = ρ L

A

LRA

ρ=

E Jρ=! !

J Eσ=! !