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Physics 202, Lecture 9

Current and Resistance (Ch 25) DC currents Ohm’s Law: Resistors and Resistance Conductivity and Resistivity

Next lecture: DC circuits

Reminder: HW #4 due tomorrow, 11 PM

Charge Motion in a Conductor

Electrons in a conductor have random motion (vave=0)

In an external electric field (e.g. as supplied by a source ofpotential difference such as a battery),electrons accelerate, produces current:

I =dQ

dt

I =!Q

!t

Average current:

Instantaneous current:

Text: 25.1direct current (DC): I constant

2

++++ +

+++

Q Q

Current: Macroscopic View

Current: rate at which chargeflows through surface:

Unit:1 Ampere = 1 C/s Current is directional: Follows positive charge (convention)+q moving in +x direction –q in moving –x direction

Charge conservation Current conservation

A

++

+ +

++

+

+

+

I

Iin IoutIin = Iout

v

Current: Microscopic ViewCurrent: motion of charged particles

Current density:(vector!)

IAnqvt

QI daverage ==

!

!=

vd: average drift velocity

n: number density

!J id!A! = I

I

J =I

A= nqvd

3

Drift velocity of conduction electrons

acceleration a=qE/mvelocity v = vo + qEt / m

Lots of particles: averageover all particle velocitiesvave = (vo)ave + qEtave / m = qEτ / m

vd= Drift velocity= qEτ / m

vo= velocity after last collisiont = time since last collision

x

x

x

x

xx

τ=ave. time since last collision

In external electric field:

Ohm’s Law: ResistanceI = current = nqvdAvd= Drift velocity= qEτ / m

�

I = nqqE!m

"

# $

%

& ' A =

nq2!m

A

L

"

# $

%

& ' V (V = IR

Voltage proportional to current! This is Ohm’s law.

�

R =nq

2!m

A

L

"

# $

%

& '

(1

!J = nq

!vd =

nq2!m

"#$

%&'!E = (

!E =

1

)

!E

conductivity

resistivity

Text: 25.6

4

Conductivity, Resistivity, Resistance Ohm’s Law (microscopic): J=σE

Ohm’s Law (macroscopic): ΔV=IR

Resistance R (unit: Ohm Ω = Volt/Ampere)Exercise: relate R to ρ

AR

!!=

Resistivity (intrinsic)

Length &Cross-section

(shape) Resistance

ResistorsResistorsResistivity For Various Materials

AR

!!=

Text: 25.11, 25.14

5

Ohmic and non-Ohmic Materials

non-Ohmic:

Nonlinear I-V relationship

Ohmic:

Linear I-V relationship

(constant resistance over wide range of voltages)

Resistance And TemperatureResistivity is usually temperature dependent.

Normal Metal

Semiconductor

ρ

T

Superconductor

6

SuperconductivitySuperconductors: temperature T<TC, resistivity ρ=0

(a quantum phenomenon!)

Electrical Power

Electrical Power:

Ohmic:

P =dU

dt=d(Q!V )

dt= I!V

P = I2R =

(!V )2

R

ΔV

I IR

+ -ΔV

Text example: 25.35(power delivered to resistor)

Unit: watts (W)energy unit: kWh

1 KWH = 3.6 MJ

7

Example: Battery Connected To A Resistor

Energy flow of this battery-resistor set-up

+

-

1.5V

-------

+++++

Res

isto

r

R

e-

I

Chemical Process ΔV =1.5VΔV on Resistor Current I= ΔV/R

Charge flow through the resistor in Δt:Q=IΔt = ΔV/RΔt

Electrical potential energy released:U=QΔV = ΔV/RΔt ΔV = (ΔV)2/RΔt

Power: P=U/dt = (ΔV)2/R

Energy Flow: Chemical Electrical U KE thermal/light

e-

motion due to chemical process

motion due to E. field collisions

-----

++++

e-

Quiz: Consumption of Electric Power On Resistors

A voltage is applied to a wire of length L . If L increases,does power consumed increase or decrease?

IncreasesDecreasesSame

Ni

ΔV

8

Quiz: Consumption of Electric Power On Resistors

When a current passes through serially connected wiresegments made of copper and nichrome, whichmetal: copper or nichrome, consumes more energy?

(ρCu ~ 10-8 Ωm, ρNi ~ 10-6 Ωm, All segments have about the samelength and diameter.)

CopperNichromeSame

Cu Ni Cu NiI

Resistors in Circuits

• Resistors:

Purpose is to limit thecurrent in a circuit.

Next lecture: DC circuits with resistors, capacitors

ΔV

I IR

+ -ΔVBasic rule: voltage “drops”

as current flows through resistor

But first: a preview of resistors in series, parallel

9

Resistors in Series: Preview

a

c

Reffective

a

b

c

R1

R2

I

Intuitively: voltage “drops”

Hence: Reffective = (R1 + R2 )

Va!V

b= IR

1

Vb!V

c= IR

2

Va!V

c= I(R

1+ R

2)

Current same in both!

Another (intuitive) way…

Consider two cylindrical resistors with lengths L1 and L2 :

V

R1

R2

L2

L1

Put them together, end to end to make a longer one...

R1= !

L1

A

R2= !

L2

A

Reffective = !L1+ L

2

A= R

1+ R

2R = R

1+ R

2

What about parallel?