2/6/07184 Lecture 171 PHY 184 Spring 2007 Lecture 17 Title: Resistance and Circuits.

2/6/07 184 Lecture 17 1

PHY 184PHY 184PHY 184PHY 184

Spring 2007Lecture 17

Title: Resistance and Circuits

2/6/07 184 Lecture 17 2

AnnouncementsAnnouncementsAnnouncementsAnnouncements

Homework Set 4 is done! Midterm 1 will take place in class on Thursday

• Chapters 16 - 19• Homework Set 1 - 4• You may bring one 8.5 x 11 inch sheet of equations, front and back,

prepared any way you prefer.• Bring a calculator• Bring a No. 2 pencil• Bring your MSU student ID card

We will post Midterm 1 as Corrections Set 1 after the exam• You can re-do all the problems in the Exam• You will receive 30% credit for the problems you missed

• To get credit, you must do all the problems in Corrections Set 1, not just the ones you missed

2/6/07 184 Lecture 17 3

Seating Instructions ThursdaySeating Instructions ThursdaySeating Instructions ThursdaySeating Instructions Thursday

Fall Semester 2006Midterm 1Section 1

Alphabetical Seating Order

Please seat yourselves alphabetically

Sit in the row (C, D,…) corresponding to your last name alphabetically

For example, Bauer would sit in row C, Westfall in Row O

We will pass out the exam by rows

Section 2

2/6/07 184 Lecture 17 4

ReviewReviewReviewReview

The property of a particular device or object that describes it ability to conduct electric currents is called the resistance, R

The definition of resistance R is

The unit of resistance is the ohm,

R V

iR

V

i

1 1 V

1 A1

1 V

1 A

2/6/07 184 Lecture 17 5

Review (2)Review (2)Review (2)Review (2) The resistance R of a device is given by

is resistivity of the material from which the device is constructed

L is the length of the device A is the cross sectional area of the device

R L

AR

L

A

2/6/07 184 Lecture 17 6

Temperature Dependence of ResistivityTemperature Dependence of ResistivityTemperature Dependence of ResistivityTemperature Dependence of Resistivity

The resistivity (and hence resistance) varies with temperature. For metals, this dependence on temperature is linear over a broad

range of temperatures. An empirical relationship for the temperature dependence of the

resistivity of metals is given by

0 0 T T0 0 0 T T0 • is the resistivity at temperature

T

• 0 is the resistivity at some standard temperature T0

• is the “temperature coefficient” of electric resistivity for the material under consideration

Copper

2/6/07 184 Lecture 17 7

Temperature Dependence of ResistanceTemperature Dependence of ResistanceTemperature Dependence of ResistanceTemperature Dependence of Resistance

In everyday applications we are interested in the temperature dependence of the resistance of various devices.

The resistance of a device depends on the length and the cross sectional area.

These quantities depend on temperature However, the temperature dependence of linear

expansion is much smaller than the temperature dependence of resistivity of a particular conductor.

So the temperature dependence of the resistance of a conductor is, to a good approximation,

R R0 R0 T T0 R R0 R0 T T0

2/6/07 184 Lecture 17 8

Temperature DependenceTemperature DependenceTemperature DependenceTemperature Dependence

Our equations for temperature dependence deal with relative temperatures so that one can use °C as well as K.

Values of for representative metals are shown below

Material Temperature Coefficien t of Resistivity, (K-1) Silver 4.110-3

Copper 4.310-3 Aluminum 4.410-3 Iron 6.510-3

2/6/07 184 Lecture 17 9

Other Temperature DependenceOther Temperature DependenceOther Temperature DependenceOther Temperature Dependence

At very low temperatures the resistivity of some materials goes to exactly zero.

These materials are called superconductors• Many applications including MRI

The resistance of some semiconducting materials actually decreases with increasing temperature.

These materials are often found in high-resolution detection devices for optical measurements or particle detectors.

These devices must be kept cold to keep their resistance high using refrigerators or liquid nitrogen.

2/6/07 184 Lecture 17 10

Ohm’s LawOhm’s LawOhm’s LawOhm’s Law

To make current flow through a resistor one must establish a potential difference across the resistor.

This potential difference is termed an electromotive force, emf.

A device that maintains a potential difference is called an emf device and does work on the charge carriers

The emf device not only produces a potential difference but supplies current.

The potential difference created by the emf device is termed Vemf .

We will assume that emf devices have terminals that we can connect and the emf device is assumed to maintain Vemf between these terminals.

2/6/07 184 Lecture 17 11

Ohm’s Law (2)Ohm’s Law (2)Ohm’s Law (2)Ohm’s Law (2)

Examples of emf devices are• Batteries that produce emf through chemical reactions• Electric generators that create emf from electromagnetic

induction• Solar cells that convert energy from the Sun to electric energy

In this chapter we will assume that the source of emf is a battery.

A circuit is an arrangement of electrical components connected together with ideal conducting wires (i.e., having no resistance).

Electrical components can be sources of emf, capacitors, resistors, or other electrical devices.

We will begin with simple circuits that consist of resistors and sources of emf.

2/6/07 184 Lecture 17 12

Ohm’s Law (3)Ohm’s Law (3)Ohm’s Law (3)Ohm’s Law (3) Consider a simple circuit of the form shown below

Here a source of emf provides a voltage V across a resistor with resistance R.

The relationship between the voltage and the resistance in this circuit is given by Ohm’s Law

… where i is the current in the circuit

Vemf iRVemf iR

( agrees with def. of R = V / i )

2/6/07 184 Lecture 17 13

Ohm’s ClickerOhm’s ClickerOhm’s ClickerOhm’s Clicker

What is the resistance of the resistor in this Demo?

A) about 1 B) about 100 C) about 10

2/6/07 184 Lecture 17 14

Ohm’s ClickerOhm’s ClickerOhm’s ClickerOhm’s Clicker

What is the resistance of the resistor in this Demo?

C) about 10

R V

iR

V

i

2/6/07 184 Lecture 17 15

Ohm’s Law (4)Ohm’s Law (4)Ohm’s Law (4)Ohm’s Law (4)

Now let’s visualize the same circuit in a different way, making it clearer where the potential drop happens and what part of the circuit is at which potential.

The top part of this drawing is just our original circuit diagram.

In the bottom part we show the same circuit, but now the vertical dimension represents the voltage drop around the circuit.

The voltage is supplied by the source of emf and the entire voltage drop occurs across the single resistor.

2/6/07 184 Lecture 17 16

Resistances in SeriesResistances in SeriesResistances in SeriesResistances in Series

Resistors connected such that all the current in a circuit must flow through each of the resistors are connected in series.

For example, two resistors R1 and R2 in series with one source of emf with voltage Vemf implies the circuit shown below

2/6/07 184 Lecture 17 17

Two Resistors in 3DTwo Resistors in 3DTwo Resistors in 3DTwo Resistors in 3D

To illustrate the voltage drops in this circuit we can represent the same circuit in three dimensions.

The voltage drop across resistor R1 is V1 .

The voltage drop across resistor R2 is V2 .

The sum of the two voltage drops must equal the voltage supplied by the battery

Vemf V1 V2

2/6/07 184 Lecture 17 18

Resistors in SeriesResistors in SeriesResistors in SeriesResistors in Series

The current must flow through all the elements of the circuit so the current flowing through each element of the circuit is the same.

For each resistor we can apply Ohm’s Law

… where

We can generalize this result to a circuit with n resistors in series

Vemf iR1 iR2 iReq

Req R1 R2

Req Rii1

n

Req Rii1

n

2/6/07 184 Lecture 17 19

Example: Internal Resistance of a BatteryExample: Internal Resistance of a BatteryExample: Internal Resistance of a BatteryExample: Internal Resistance of a Battery

When a battery is not connected in a circuit, the voltage across its terminals is Vt

When the battery is connected in series with a resistor with resistance R, current i flows through the circuit.

When current is flowing, the voltage, V, across the terminals of the battery is lower than Vt .

This drop occurs because the battery has an internal resistance, Ri, that can be thought of as being in series with the external resistor.

We can express this relationship as

Vt iReq i R Ri

2/6/07 184 Lecture 17 20

Example: Internal Resistance of a Battery Example: Internal Resistance of a Battery (2)(2)


We can represent the battery, its internal resistance and the external resistance in this circuit diagram

battery

terminals of the battery

Consider a battery that has a voltage of 12.0 V when it is not connected to a circuit.

When we connect a 10.0 resistor across the terminals, the voltage across the battery drops to 10.9 V.

What is the internal resistance of the battery?

2/6/07 184 Lecture 17 21



The current flowing through the external resistor is

The current flowing in the complete circuit must be the same so

i V

R

10.9 V

10.0 1.09 A

Vt iReq i R Ri

R Ri Vi

Ri V

i R

12.0 V

1.09 A 10.0 1.0

2/6/07 184 Lecture 17 22

Resistances in ParallelResistances in ParallelResistances in ParallelResistances in Parallel

Instead of connecting resistors in series so that all the current must pass through both resistors, we can connect the resistors in parallel such that the current is divided between the two resistors.

This type of circuit is shown is below

2/6/07 184 Lecture 17 23

Resistance in Parallel (2)Resistance in Parallel (2)Resistance in Parallel (2)Resistance in Parallel (2)

In this case the voltage drop across each resistor is equal to the voltage provides by the source of emf.

Using Ohm’s Law we can write the current in each resistor

The total current in the circuit must equal the sum of these currents

Which we can rewrite as

i1 VemfR1

i2 VemfR2

i i1 i2

i i1 i2 VemfR1

VemfR2

Vemf1

R1

1

R2

2/6/07 184 Lecture 17 24

Resistance in Parallel (3)Resistance in Parallel (3)Resistance in Parallel (3)Resistance in Parallel (3)

We can then rewrite Ohm’s Law for the complete circuit as

.. where

We can generalize this result for two parallel resistors to n parallel resistors

i Vemf1

Req

1

Req

1

R1

1

R2

1

Req

1

Rii1

n

1

Req

1

Rii1

n

2/6/07 184 Lecture 17 25

Clicker QuestionClicker QuestionClicker QuestionClicker Question

A battery, with potential V across it, is connected to a combination of two identical resistors and then has a current i through it. What are the potential differences V across and the current through either resistor if the two resistors are in series?

A) V, 2i B) V, i/2 C) V/2, i

2/6/07 184 Lecture 17 26


A battery, with potential V across it, is connected to a combination of two identical resistors and then has a current i through it. What are the potential differences V across and the current through either resistor if the two resistors are in series?

C) V/2, i

In series: The resistors have identical currents i The sum of potential differences across the resistors is

equal to the applied potential difference:

2/6/07 184 Lecture 17 27


A battery, with potential V across it, is connected to a combination of two identical resistors and then has a current i through it. What are the potential differences V across and the current through either resistor if the two resistors are in parallel?

A) V, 2i B) V, i/2 C) 2V, i

2/6/07 184 Lecture 17 28


A battery, with potential V across it, is connected to a combination of two identical resistors and then has a current i through it. What are the potential differences V across and the current through either resistor if the two resistors are in parallel?

B) V, i/2

In parallel: The resistors all have the same V applied The sum of the currents through the resistors is equal

to the total current:

i i1 i2

2/6/07184 Lecture 171 PHY 184 Spring 2007 Lecture 17 Title: Resistance and Circuits.

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