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Applied Physics

Lecture 8Lecture 8 Electrodynamics

Electric current current and drift speed resistance and Ohm’s law resistivity temperature variation of resistance electrical energy and power

Chapter 17

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Lightning ReviewLightning Review

Last lecture:

1.1. Capacitance and capacitorsCapacitance and capacitors Capacitors with dielectrics (CCapacitors with dielectrics (C↑ if ↑ if ↑↑))

2.2. Current and resistanceCurrent and resistance Electric currentElectric current Current and drift speedCurrent and drift speed

Review Problem: A parallel-plate capacitor is attached to a battery that maintains a constant potential difference V between the plates. While the battery is still connected, a glass slab is inserted so as to just fill the space between the plates. The stored energy

a. increasesb. decreases c. remains the same

0 0,A

C C Cd

QI

t

dI nqv A

221 1

2 2 2

QU QV CV

C

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15.2 Current and Drift Speed15.2 Current and Drift Speed

Consider the current on a conductor of cross-sectional Consider the current on a conductor of cross-sectional area A. area A.

Avdq

vdt

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15.2 Current and Drift Speed (2)15.2 Current and Drift Speed (2)

Volume of an element of length Volume of an element of length x is : x is : V = A V = A x.x.

Let n be the number of carriers per unit of volume.Let n be the number of carriers per unit of volume.

The total number of carriers in The total number of carriers in V is: n A V is: n A x.x.

The charge in this volume is: The charge in this volume is: Q = (n A Q = (n A x)q.x)q.

Distance traveled at Distance traveled at drift speed drift speed vvdd by carrier in time by carrier in time t: t: x = x = vvd d t.t.

Hence: Hence: Q = (n A Q = (n A vvd d t)q.t)q.

The current through the conductor: The current through the conductor:

I = I = Q/ Q/ t = n A t = n A vvd d qq..

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15.2 Current and Drift Speed (3)

• In an isolated conductor, charge carriers move randomly in all directions.

• When an external potential is applied across the conductor, it creates an electric field inside which produces a force on the electron.

• Electrons however still have quite a random path.

• As they travel through the material, electrons collide with other electrons, and nuclei, thereby losing or gaining energy.

• The work done by the field exceeds the loss by collisions.

• The electrons then tend to drift preferentially in one direction.

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15.2 Current and Drift Speed - Example15.2 Current and Drift Speed - Example

Question:Question:

A copper wire of cross-sectional area 3.00x10A copper wire of cross-sectional area 3.00x10-6-6 m m22 carries a current of carries a current of 10. A. Assuming that each copper atom contributes one free electron to 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 the metal, find the drift speed of the electron in this wire. The density of copper is 8.95 g/cmcopper is 8.95 g/cm33..

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Question:

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.

Reasoning: We know:• A = 3.00x10-6 m2 • I = 10 A.• = 8.95 g/cm3.• q = 1.6 x 10-19 C.• n = 6.02x1023 atom/mol x 8.95 g/cm3 x ( 63.5 g/mol)-1

• n = 8.48 x 1022 electrons/ cm3.

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Question:

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.

Ingredients:

A = 3.00x10-6 m2 ; I = 10 A.; = 8.95 g/cm3.; q = 1.6 x 10-19 C.

n = 8.48 x 1022 electrons/ 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

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15.2 Current and Drift Speed - Comments15.2 Current and Drift Speed - Comments

Drift speeds are usually Drift speeds are usually very smallvery small..

Drift speed much smaller than the average speed Drift speed much smaller than the average speed between collisions. between collisions.

Electrons traveling at 2.46x10Electrons traveling at 2.46x10-6-6 m/s would would take 68 min to m/s would would take 68 min to travel 1m.travel 1m.

So why does light turn on so quickly when one flips a So why does light turn on so quickly when one flips a switch?switch?

The info (electric field) travels at roughly 10The info (electric field) travels at roughly 1088 m/s… m/s…

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Mini-quizMini-quiz

Consider a wire has a long conical shape. How does the Consider a wire has a long conical shape. How does the velocity of the electrons vary along the wire?velocity of the electrons vary along the wire?

Every portion of the wire carries the same current: as the cross Every portion of the wire carries the same current: as the cross sectional area decreases, the drift velocity must increase to sectional area decreases, the drift velocity must increase to carry the same value of current. This is due to the electrical field carry the same value of current. This is due to the electrical field lines being compressed into a smaller area, thereby increasing lines being compressed into a smaller area, thereby increasing the strength of the electric field.the strength of the electric field.

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17.3 Resistance and Ohm’s Law - Intro17.3 Resistance and Ohm’s Law - Intro

When a voltage (potential difference) is applied across When a voltage (potential difference) is applied across the ends of a metallic conductor, the current is found to the ends of a metallic conductor, the current is found to be proportional to the applied voltage.be proportional to the applied voltage.

I V V

I

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17.3 Definition of Resistance17.3 Definition of Resistance

In situations where the proportionality is exact, one can In situations where the proportionality is exact, one can write.write.

V IR • The proportionality constant R is called resistance

of the conductor.• The resistance is defined as the ratio.

VR

I

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17.3 Resistance - Units17.3 Resistance - Units

In SI, resistance is expressed in In SI, resistance is expressed in volts per amperevolts per ampere..

A special name is given: A special name is given: ohms (ohms ().).

Example: if a potential difference of 10 V applied across Example: if a potential difference of 10 V applied across a conductor produces a 0.2 A current, then one a conductor produces a 0.2 A current, then one concludes the conductors has a resistance of 10 V/0.2 a concludes the conductors has a resistance of 10 V/0.2 a = = 50 50 ..

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17.3 Ohm’s Law17.3 Ohm’s Law

Resistance in a conductor Resistance in a conductor arises because of collisions arises because of collisions between electrons and fixed charges within the materialbetween electrons and fixed charges within the material ..

In many materials, including most metals, the resistance is In many materials, including most metals, the resistance is constantconstant over a wide range of applied voltages. over a wide range of applied voltages.

This is a statement of Ohm’s law.This is a statement of Ohm’s law.

Georg Simon Ohm(1787-1854)

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I

V

I

V

Linear or Ohmic MaterialNon-Linear or Non-Ohmic Material

Semiconductorse.g. devices called diodes

Most metals, ceramics

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V IR

Ohm’s Law

R understood to be independent of V.

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Definition: resistorDefinition: resistor

Resistor: a conductor that provides a specified resistance in an electric circuit.

I

V = IR

+ -

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Example: Example: Resistance of a Steam IronResistance of a Steam Iron

All household electric devices are required to have a All household electric devices are required to have a specified resistance (as well as many other specified resistance (as well as many other characteristics…). Consider that the plate of a certain characteristics…). Consider that the plate of a certain steam iron states the iron carries a current of 7.40 A when steam iron states the iron carries a current of 7.40 A when connected to a 120 V source. What is the resistance of the connected to a 120 V source. What is the resistance of the steam iron?steam iron?

12016.2

7.40

V VR

I A

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17.4 Resistivity - Intro17.4 Resistivity - Intro

Electrons moving inside a conductor subject to an Electrons moving inside a conductor subject to an external potential constantly collide with atoms of the external potential constantly collide with atoms of the conductor.conductor.

They lose energy and are repeated re-accelerated by the They lose energy and are repeated re-accelerated by the electric field produced by the external potential.electric field produced by the external potential.

The collision process is equivalent to an internal friction.The collision process is equivalent to an internal friction.

This is the origin of a material’s This is the origin of a material’s resistanceresistance..

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The resistance of an ohmic conductor is proportional to The resistance of an ohmic conductor is proportional to the its length, the its length, ll, and inversely proportional to the cross , and inversely proportional to the cross section area, section area, AA, of the conductor., of the conductor.

17.4 Resistivity - Definition17.4 Resistivity - Definition

lR

A

• The constant of proportionality is called the resistivity of the material.

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17.4 Resistivity - Remarks17.4 Resistivity - Remarks

Every material has a characteristic resistivity that Every material has a characteristic resistivity that depends on its electronic structure, and the temperature.depends on its electronic structure, and the temperature.

Good Good conductorsconductors have have low resistivitylow resistivity..

InsulatorsInsulators have have high resistivityhigh resistivity..

Analogy to the flow of water through a pipe.Analogy to the flow of water through a pipe.

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17.4 Resistivity - Units17.4 Resistivity - Units

Resistance expressed in Ohms, Resistance expressed in Ohms,

Length in meter.Length in meter.

Area are mArea are m22, ,

Resistivity thus has units of Resistivity thus has units of mm..

lR

A RA

l

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Material Resistivity (10-8  m) Material Resistivity (10-8  m)

Silver 1.61 Bismuth 106.8

Copper 1.70 Plutonium 141.4

Gold 2.20 Graphite 1375

Aluminum 2.65 Germanium 4.6x107

Pure Silicon

3.5 Diamond 2.7x109

Calcium 3.91 Deionized water

1.8x1013

Sodium 4.75 Iodine 1.3x1015

Tungsten 5.3 Phosphorus 1x1017

Brass 7.0 Quartz 1x1021

Uranium 30.0 Alumina 1x1022

Mercury 98.4 Sulfur 2x1023

Resistivity of various materials

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Mini-quizMini-quizWhy do old light bulbs give less light than when Why do old light bulbs give less light than when new?new?

Answer:

• 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.

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17.4 Resistivity - Example(a) Calculate the resistance per unit length of a 22-gauge nichrome wire of radius 0.321 m.

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

Resistivity (Table): 1.5 x 10 m.

6

7 2

1.5 104.6

3.24 10m

R m

l A m

Resistance/unit length:

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17.4 Resistivity - Example

(b) If a potential difference of 10.0 V is maintained across a 1.0-m length of the nichrome wire, what is the current?

10.02.2

4.6

V VI A

R

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17.4 Temperature Variation of Resistance - Intro• The resistivity of a metal depends on many

(environmental) factors.• The most important factor is the temperature.• For most metals, the resistivity increases with

increasing temperature.• The increased resistivity arises because of larger

friction caused by the more violent motion of the atoms of the metal.

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For most metals, resistivity increases For most metals, resistivity increases approx. linearly with temperature.approx. linearly with temperature.

1o oT T

• is the resistivity at temperature T (measured in Celsius).

• is the reference resistivity at the reference temperature T (usually taken to be 20 oC).

• is a parameter called temperature coefficient of resistivity.

For a conductor with fixed cross section.For a conductor with fixed cross section.

1o oR R T T

TMetallic Conductor

TSuperconductor

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17.5 Temperature Variation of Resistance - 17.5 Temperature Variation of Resistance - ExampleExample

Platinum Resistance ThermometerPlatinum Resistance Thermometer

A resistance thermometer, which measures temperature by measuring the A resistance thermometer, which measures temperature by measuring the change in the resistance of a conductor, is made of platinum and has a change in the resistance of a conductor, is made of platinum and has a resistance of 50.0 resistance of 50.0 at 20 at 20ooC. When the device is immersed in a vessel C. When the device is immersed in a vessel containing melting indium, its resistance increases to 76.8 containing melting indium, its resistance increases to 76.8 . Find the melting . Find the melting point of Indium.point of Indium.

Solution:

Using =3.92x10-3(oC)-1 from table 17.1.

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Platinum Resistance ThermometerPlatinum Resistance ThermometerA resistance thermometer, which measures temperature by measuring the change in the A resistance thermometer, which measures temperature by measuring the change in the resistance of a conductor, is made of platinum and has a resistance of 50.0 resistance of a conductor, is made of platinum and has a resistance of 50.0 at 20 at 20ooC. C. When the device is immersed in a vessel containing melting indium, its resistance When the device is immersed in a vessel containing melting indium, its resistance increases to 76.8 increases to 76.8 . Find the melting point of Indium.. Find the melting point of Indium.

Solution:Solution:

Using Using =3.92x10=3.92x10-3-3((ooC)C)-1-1 from table 17.1. from table 17.1.

RRoo=50.0 =50.0 ..

TToo=20=20ooC.C.

R=76.8 R=76.8 .. 13

76.8 50.0

3.92 10 50.0

137

157

oo

oo

o

o

R RT T

R C

C

T C