Chapter 27 Current and Resistance

• Scalar

• Sense determined by the movement of the positive charge carrier

Average Electric CurrentQ

It

Instantaneous Electric Current dQI

dt

Units:Coulomb C

Ampere Asec s

Microscopic view of current

+++

+

++

E

J

Isn’t E = 0?

Current density, j, is the electric current per unit cross sectional area. Current density is a vector.

Charge carriers experience a force and accelerate. They collide with the atoms in the metal and are slowed down eventually reaching a terminal velocity called the drift velocity

IJ

A

A

Resistance model – Why doesn’t the electron accelerate?

e e

qE eEa

m m

f i ie

eEv v at v t

m

f de

eEv v

m

Counting the charge flow

+++

+

++

E

j

dv

A

Let: n number of charge carriers per unit volume

A cross sectional area of the wire

q charge on a charge carrier (-e)

dv t

dd

nA v t qQI nA v q

t t

dQ nA v t q

Current density for electrons

---

-

--

E

J

dd

nA v qIJ n v q

A A

A

dJ nv e

dv

I J A

For non-uniformcurrent density

I J dA

E

vd

Ohm’s Law

+++

+

++

E

J

J E

2

e

neconductivity

m

e2

mresistivity

ne

J E

E J

1

2

de e

eE neJ nv e n e E

m m

Scalar form of Ohm’s Law

b

ba

a

IV E ds E J

A

+++

+

++

E

J

b

a

E J

baV I IRA

R

A A

Units of resistance, resistivity, and conductivity

RA A

Ohms

RA

1

RA

Ohms meter m

11 mho siemens

mm m m

Temperature effect on resistivity

o o(T) 1 T T

o

1= temperature coefficient of resistivity

T

o oR(T) R 1 T T

RA

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

Superconductor Application

• An important application of superconductors is a superconducting magnet

• The magnitude of the magnetic field is about 10 times greater than a normal electromagnet

• Used in MRI units

Electrical energy and power

b aba

U U UV

q q

V

dU Vdq

dU dqP V IV

dt dt

dUV

dq

The power transformed in an electric device is then:

V IRUsing Ohm’s Law 2P IV I IR I R

2V VP IV V

R R

Example Problem

Given: Copper wire transmission linelength = 1500 mdiameter = 0.1cmcurrent = 50 Acu = 1.67 x 10-6 -cm

Find: Power loss due to heating the wire

Example P27.6

The quantity of charge q (in coulombs) that has passed through a surface of area 2.00 cm2 varies with time according to the equation q = 4t3 + 5t + 6, where t is in seconds.

(a) What is the instantaneous current through the surface at t = 1.00 s?

(b) What is the value of the current density?

2

1.00 s1.00 s

1.00 s 12 5 17.0 Att

dqI t

dt

2

4 2

17.0 A85.0 kA m

2.00 10 m

IA

J

Example P27.15

A 0.900-V potential difference is maintained across a 1.50-m length of tungsten wire that has a cross-sectional area of 0.600 mm2. What is the current in the wire?

2

2 7 21.00 m0.600 mm 6.00 10 m

1000 mmA

7 2

8

0.900 V 6.00 10 m

5.60 10 m 1.50 m

VAI

6.43 AI

Example P27.29

A certain lightbulb has a tungsten filament with a resistance of 19.0 Ω when cold and 140 Ω when hot. Assume that the resistivity of tungsten varies linearly with temperature even over the large temperature range involved here, and find the temperature of the hot filament. Assume the initial temperature is 20.0°C.

3140 19.0 1 4.50 10 C T

31.42 10 C 20.0 CT T

31.44 10 CT

Resistor Values

• Values of resistors are commonly marked by colored bands

Alternating Current

oV t V

oV t V sin t

oo

V t VI(t) sin t I sin t

R R

oI t I

AC Power

o

2 2 2P I R I R sin t

22 oo

V1 1P I R

2 2 R

T / 2

2

T / 2

1 1sin t dt

T 2

?

Root Mean Square (rms)

oV t V sin t oI(t) I sin t

22 oV

V2

22 o o

rms

V VV V

2 2

22 oI

I2

22 o o

rms

I II I

2 2

2 2o rms

1P I R I R

2

2 2o rmsV V1

P2 R R

Electrical Safety

• Current kills, not voltage (70 mA)• Normal body resistance = 105

But could be less than 1000

• Take advantage of insulators, remove conductors

• Work with one hand at a time• Shipboard is more dangerous• Electrical safety is an officer responsibility