Equipment for Electricity
Figure 1:
Equipment for Electricity
balance
copper wire
glass rod
heater wire
multimeter
ohmmeter
power supply
resistors
ruler
wire cutters


  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.

      In this lesson we will begin our discussion of electricity itself. In the simplest microscope model of conduction in a metal, each atom in the crystal lattice gives up one or more of its outer electrons. These electrons are then free to move through the crystal lattice, colliding with stationary positive ions. If there is no electric field, the electrons move in straight lines between collisions, the direction of their velocities is 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...]


Start-up questions

  1. What parameters of a material determine its electrical resistance?
  2. Do different materials have different resistance?
  3. What is electrical resistance?
  4. Do different shapes of the same material have different resistance values?

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 ohmís 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.

     r (resistivity, unit of W·m)     (This is independent of shape, intrinsic property)
     s (conductivity, unit of 1/W·m)     (this is independent of shape also, intrinsic)
     R (resistance, unit of W) = V/I = r (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 = r (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.

Resistivities at Room Temperature

Metal

r (W·m)  

Substance

r (W·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 =

      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 + aT)

           where   Ro = resistance at 0°C
                                    R = resistance at T°C
                                    a = 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 W·m
    Resistance R = r × L/A = 1.72×10-8 W·m × 30.0 m / (p × 1.025² mm²)
                         R = 1.72×10-8 W·m × 30 m / (p × 0.001025² m²)
                         R = 0.156 W
  3. If I have a heater wire which has a resistance of 100. W at 300.°C and a resistance of 120. W at 800.°C. What is its resistance at 2.00×10³ °C?

    Solution: R = Ro(1 + aT) where T is temperature in °C.
    Substituting the two conditions into the equation, we have

    [1]  100. W = Ro (1 + 300.°C a) or 100. W = Ro + 300.°C aRo

    [2]  120. W = Ro (1 + 800.°C a) or 120. W = Ro + 800.°C aRo

    Subtract equation [1] from [2] we have:
          20. W = 500.°C aRo
              so aRo = 0.040 W/°C   [3]
    Substituting aRo = 0.040 W/°C into equation [1] we have
               100. W = Ro+300.°C × 0.040 W/°C    or   Ro = 88 W   [4]
    Finally substituting aRo = 0.040 W/°C and Ro = 88 W into the resistance equation:
               R = Ro(1 + aT) = Ro + aRoT = 88 W + 0.040 W/°C × 2000°C
    We have R = 170 W.

Homework Only -- for Printing Solution

Homework
  1. a. What length of copper wire 0.600 mm in diameter has a resistance of 100. W?
    b. What length of glass wire 0.600 mm in diameter has a resistance of 100. W?

  2. I have an electrical wire with an "a" value (in the temperature dependent equation) of 2.00×10-4/°C and its resistance at room temperature (25.0°C) is 200. W. What is its resistance at 2.00×10³ °C?