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PHY 184. Spring 2007 Lecture 17. Title: Resistance and Circuits. Announcements. 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.

PHY 184

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## PHY 184

Spring 2007

Lecture 17

Title: Resistance and Circuits

184 Lecture 17

### Announcements

• 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

184 Lecture 17

Section 2

## Seating Instructions Thursday

• 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

Fall Semester 2006

Midterm 1

Section 1

Alphabetical Seating Order

184 Lecture 17

### Review

• 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, 

184 Lecture 17

### 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

184 Lecture 17

### Temperature 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

Copper

• is the resistivity at temperature T

• 0is the resistivity at some standard temperature T0

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

184 Lecture 17

### Temperature 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,

184 Lecture 17

### Temperature 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

184 Lecture 17

### Other 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.

184 Lecture 17

### Ohm’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.

184 Lecture 17

### 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.

184 Lecture 17

### 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

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

184 Lecture 17

### Ohm’s Clicker

• What is the resistance of the resistor in this Demo?

184 Lecture 17

### Ohm’s Clicker

• What is the resistance of the resistor in this Demo?

184 Lecture 17

### 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.

184 Lecture 17

### Resistances 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

184 Lecture 17

### Two 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

184 Lecture 17

### Resistors 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

184 Lecture 17

### Example: 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

184 Lecture 17

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

terminals of the battery

battery

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

• 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?

184 Lecture 17

### Example: Internal Resistance of a Battery (3)

• The current flowing through the external resistor is

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

184 Lecture 17

### Resistances 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

184 Lecture 17

### 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

184 Lecture 17

### 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

184 Lecture 17

### Clicker 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

184 Lecture 17

### Clicker 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?

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:

184 Lecture 17

### Clicker 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 parallel?

A) V, 2i

B) V, i/2

C) 2V, i

184 Lecture 17

### Clicker 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 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:

184 Lecture 17