Electricity
After the exploration of a mechanical-mechanical
energy conversion in the last lesson and covering
energy conversion in some form in each of the last three
lessons, the general idea of energy conversion should be
familiar to you. For much of what we do with energy in
our homes, we start with electricity and convert it to
other forms of energy.
Figure 1:
Equipment for Electricity
In this lesson we will begin our discussion of
balance
electricity itself. In the simplest microscope model of
copper wire
glass rod
conduction in a metal, each atom in the crystal lattice
heater
wire
gives up one or more of its outer electrons. These
multimeter
electrons are then free to move through the crystal
ohmmeter
lattice, colliding with stationary positive ions. If there is
power supply
no electric field, the electrons move in straight lines
resistors
ruler
between collisions, the direction of their velocities is
wire
cutters
random, and on average they never get anywhere. But if
an electric field is present, the paths curve slightly because of the acceleration caused by electric
field forces (see the description in the handout).
Objectives [At the end of this lesson students will be able to...]
•
•
•
correctly state ohm's law relating electrical current, potential and resistance.
measure resistance with a simple ohmmeter.
use a power supply and measure current with an ampere meter.
Start-up questions
1.
2.
3.
4.
What parameters of a material determine its electrical resistance?
Do different materials have different resistance?
What is electrical resistance?
Do different shapes of the same material have different resistance values?
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Electricity -- Page 1 of 6
Electricity and Ohm's Law
You have almost certainly encountered ohm's law at some point in your life. Do you recall
what it says? It is one equation that relates three terms, and can be written in three equivalent
ways:
V=IR
or I=V/R
or R=V/I
Can we use a normal resistor to verify ohm’s law? We will certainly try. We will pass current
through a resistor and measure all three terms.
Voltage across the resistor =
Current through the resistor =
Resistance value =
What happens to the resistor when I pass current through it?
Now we will discuss the factors that contribute to the resistance of a long wire. These are the
diameter, length and the material that make up the wire. The diameter and length are extrinsic
properties because they can be changed, while the material that makes up the wire itself has
intrinsic properties.
There are four terms used in describing how these factors contribute to the resistance of a
wire. They are electrical conductivity, electrical resistivity, electrical conductance and electrical
resistance.
ρ (resistivity, unit of Ω·m) (This is independent of shape, intrinsic property)
σ (conductivity, unit of 1/Ω·m) (this is independent of shape also, intrinsic)
R (resistance, unit of Ω) = V/I = ρ (L/A) (depends on shape, extrinsic)
C (conductance, unit of mho) = 1/R (depends on shape, extrinsic)
Among the description of these relationships above was a geometric connection between
resistivity and resistance:
R = ρ (L/A)
This is an important point. It reveals the direct relationship between resistivity and resistance, so
that for two "wires" of equal dimensions, the difference in resistance will be determined
completely by the intrinsic resistivities of the materials making up the two wires. This will be
illustrated later with two different materials, copper and glass.
Let's look at the resistance found in wires that deliver electricity to our homes. Copper is
usually used for wiring in houses. Using the following chart, compare the resistance (R) and
conductance (1/R) of copper with steel, aluminum, glass, and wood. Glass has a very high
resistance and therefore is used as an insulator.
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Electricity -- Page 2 of 6
Resistivities at Room Temperature
Metal
ρ (Ω
Ω·m)
Substance
ρ (Ω
Ω·m)
Silver
1.47×10-8
Pure Carbon
3.5×10-5
Copper
1.72×10-8
Amber
5×1014
Gold
2.44×10-8
Glass
1010 - 1014
Aluminum
2.63×10-8
Lucite
>1013
Steel
20×10-8
Mica
1011 - 1015
Lead
22×10-8
Quartz
75×1016
Mercury
95×10-8
Sulfur
1015
Teflon
>1013
Wood
108 - 1011
We will illustrate the concept of resistivity and resistance through the use of wires of
different composition and size. We will use copper and glass to illustrate the effect of resistivity,
and measure the resistivity of two different materials in class.
Material 1:
Length =
Cross sectional area =
Electrical current =
Voltage drop =
Resistance =
Resistivity =
Material 2:
Length =
Cross sectional area =
Electrical current =
Voltage drop =
Resistance =
Resistivity =
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Electricity -- Page 3 of 6
Any change in physical condition may give a body a new resistance different from its
previous value. Impurities, change in temperature, changes in hardness, and mechanical strains
affect this. The relationship between resistance and temperature is shown in the following
formula:
R = Ro(1 + αT)
where Ro = resistance at 0°C
R = resistance at T°C
α = constant for the material
T = temperature in °C
Is there a temperature dependence for resistivity? In an ideal crystal lattice with no atoms out
of place, a correct quantum-mechanical analysis would let the free electrons move through the
lattice with no collisions at all. But the atoms vibrate about their equilibrium positions. As
temperature increases the amplitude of these vibrations increase, and collisions become more
frequent. Therefore resistivity of a metal increases with temperature.
Assessment questions
1. What does the temperature dependence for resistivity mean for wires carrying current in
your home?
2. Why do we need circuit breakers?
Example problems
1. Why do wires carrying a current get warm?
Solution: Electrons gain energy between collisions through the work done on them by the
electric field. During collisions they transfer some of this energy to the atoms of the
material of the conductor. This leads to an increase in internal energy and temperature of
the material.
2. In household wiring, a copper wire commonly known as 12-gauge is often used. Its
diameter is 2.05 mm. Find the resistance of a 30.0 m length of this wire.
Solution: Resistivity of copper is 1.72×10-8 Ω·m
Resistance R = ρ × L/A = 1.72×10-8 Ω·m × 30.0 m / (π × 1.025² mm²)
R = 1.72×10-8 Ω·m × 30 m / (π × 0.001025² m²)
R = 0.156 Ω
3. If I have a heater wire which has a resistance of 100. Ω at 300.°C and a resistance of 120.
Ω at 800.°C. What is its resistance at 2.00×10³ °C?
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Electricity -- Page 4 of 6
Solution: R = Ro(1 + αT) where T is temperature in °C.
Substituting the two conditions into the equation, we have
[1] 100. Ω = Ro (1 + 300.°C α) or 100. Ω = Ro + 300.°C αRo
[2] 120. Ω = Ro (1 + 800.°C α) or 120. Ω = Ro + 800.°C αRo
Subtract equation [1] from [2] we have:
20. Ω = 500.°C αRo
so αRo = 0.040 Ω/°C [3]
Substituting αRo = 0.040 Ω/°C into equation [1] we have
100. Ω = Ro+300.°C × 0.040 Ω/°C or Ro = 88 Ω [4]
Finally substituting αRo = 0.040 Ω/°C and Ro = 88 Ω into the resistance equation:
R = Ro(1 + αT) = Ro + αRoT = 88 Ω + 0.040 Ω/°C × 2000°C
We have R = 170 Ω.
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Electricity -- Page 5 of 6
Homework
1. a. What length of copper wire 0.600 mm in diameter has a resistance of 100. Ω?
b. What length of glass wire 0.600 mm in diameter has a resistance of 100. Ω?
2. I have an electrical wire with an "α" value (in the temperature dependent equation) of
2.00×10-4/°C and its resistance at room temperature (25.0°C) is 200. Ω. What is its
resistance at 2.00×10³ °C?
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Electricity -- Page 6 of 6