Chapter 25Current, Resistance, Electromotive Force

• Consider current and current density

• Study the intrinsic property of resistivity

• Use Ohm’s Law and study resistance and resistors

• Connect circuits and find emf

• Examine circuits and determine the energy and power in them

• Describe the conduction of metals microscopically, on an atomic scale

1

The direction of current flow– In the absence of an external field, electrons move randomly in

a conductor. If a field exists near the conductor, its force on the electron imposes a drift.

-The electrons move at a random velocity and collide with stationary ions. Velocity in the order of 106 m/s

-Drift velocity is approximately 10-4 m/s

Current flowing– Positive charges would move with the electric field, electrons move in

opposition.– The motion of electrons in a wire is analogous to water coursing

through a river.

Chapter 25 4

Electric Current

Conventional Current Direction

Electrical current (I) in amperes is defined as the rate of electric charge flow in coulombs per second. 1 ampere (A) of current is a rate of charge flow of 1 coulomb/second.

dt

dQI (25-1)

1 mA (milliampere) = 1 x 10-3 A (ampere)

1 A(microampere) = 1 x 10-6 A (ampere)

Chapter 25 5

Electric Current Density

AdtnqvdtnAvqdQ dd )(

where n = charge carriers per unit volume q = charge per charge carrier in coulombs vd = average drift velocity of charge carriers in meters per second

Current, Drift Velocity, and Current Density

A

IJ = current density in amperes/m2

InqAvdt

dQd dnqAv

dt

dQI amperes

Chapter 25 6

Resistivity

Evd

where = mobility of conducting material

Drift Velocity is 1010 slower than Random Velocity

1 1 E

nq J

nqwhere conductivity of the material.

EEnqvnqJ d

Definition of resistivity in ohm-meters (-m).

Drift Velocity

E

J

Resistivity of the material.

Resistivity is intrinsic to a metal sample (like density is)

Resistivity and Temperature

• In metals, increasing temperature increases ion vibration amplitudes, increasing collisions and reducing current flow. This produces a positive temperature coefficient.

• In semiconductors, increasing temperature “shakes loose” more electrons, increasing mobility and increasing current flow. This produces a negative temperature coefficient.

• Superconductors, behave like metals until a phase transition temperature is reached. At lower temperatures R=0.

9

Resistance DefinedEEJ

1

EJ1

A

IJ

L

VE

L

V

A

I

1

( )I L

V L I RIA A

Ohm’s Law

where R is the resistance of the material in ohms ()

for a uniform E

+

–

V RItherefore

Solve for V

Ohm’s law an idealized model• If current density J is nearly proportional to electric field E

ratio E/J = constant and Ohm’s law applies V = I R

• Ohm’s Law is linear, but current flow through other devices may not be.

Linear Nonlinear Nonlinear

VR

I1

R

Slope1

RIV Ohm’s law applies

Resistors are color-coded for assembly work

Examples:Brown-Black-Red-Gold = 1000 ohms +5% to -5%Yellow-Violet-Orange-Silver = 47000 ohms +10% to -10%

Electromotive force and circuits

If an electric field is produced in a conductor without a complete circuit, current flows for only a very short time.

An external source is needed to produce a net electric field in a conductor. This source is an electromotive force, emf , “ee-em-eff”, (1V = 1 J/C)

Ideal diagrams of “open” and “complete” circuits

Symbols for circuit diagrams– Shorthand symbols are in use for all wiring components

15

Electromotive Force and Circuits

Electromotive Force (EMF)

EMF R

Ideal source of electrical energy

I

+

–

+VR

–

Ideal Source

Complete path needed forcurrent (I) to flow

Voltage rise in current direction

Voltage drop in current direction

Real Source

EMF

rs

R

+

–

a

b

Vab

+

–

I

Real source of electrical energy

Internal source resistance

VR = EMF = R I

R

EMF

R

VI R

External resistance

IRIrEMFV sab

Chapter 25 16

A Source with an Open Circuit

Example 25-5

VrVIrEMFVab 12012

Figure 25-16

I = 0 amps

17

A source in a complete circuitExample 25-6

Figure 25-17

IRIrVab

)( rRIIrIR

ArR

I 224

12

VIrVab 8)2(212 VIRVV baab 8)4(2''

Chapter 25 18

A Source with a Short CircuitExample 25-8

Figure 25-19

0)0( IIRIrVab 0 Ir

AV

rI 6

2

12

Ir

0abV

I = 6 A

19

Potential Rises and Drops in a Circuit

Figure 25-21

20

Energy and Power

dt

dQI

IdtdQ

dQ

dWV abab

dQVdW abab

IdtVdW abab

IVdt

dWP ab

ab watts

1 watt = 1 joule/sec

Pure Resistance

dt

dWP ab

Substitute for IdtdQ

Defined

Divide by dt

Chapter 25 21

Power Output of an EMF Source

EMF

rs

R

I

+

––

+

Vab

a

b

IRIrEMFV sab

RIrIIEMFIIrEMFIVP ssabab22)()(

Power dissipated in battery resistancePower supplied by the battery

Power dissipated in R

RIrIIEMF s22)(

+ –

Power output of battery

22

Power Input to a SourceI

sab IrEMFV

ssabab rIIEMFIIrEMFIVP 2)()(

Power dissipated in battery resistancePower charging the battery

sab rIIEMFIV 2)(

a

b

Total Power input to battery

Vab greater then the EMF of the battery+

–

+–+

–

23

Power Input and Output in a Complete CircuitExample 25-9

Figure 25-25

24

Power in a Short Circuit

Example 25-11

Theory of Metallic Conduction

• Simple, non-quantum-mechanical model

• Each atom in a metal crystal gives up one or more electrons that are free to move in the crystal.

• The electrons move at a random velocity and collide with stationary ions. Velocity in the order of 106 m/s (drift velocity is approximately 10-4 m/s)

• The average time between collisions is the mean free time, τ.

• As temperature increases the ions vibrate more and produce more collisions, reducing τ.

Chapter 25 25

A microscopic look at conduction

– Consider Figure 25.27.– Consider Figure 25.28.– Follow Example 25.12.