Peter's Physics Pages
Peter's Index Physics Home Course Index Lecture 2
Physics for Civil Engineering
This is an introduction to Electricity, Strength of Materials and Waves.
Lecture 1 (Current, Potential Difference and Resistance)
In this lecture the following are introduced:
• Electric Charge, Current and Potential Difference
• Microscopic picture of current in a wire
• Conductors, Conductance and Conductivity
• Resistors, Resistance and Resistivity
• Conductors, Insulators and Semiconductors
• Temperature coefficient of resistivity
• Heating effect of current
Electric charge
From people like Thales, it was known around 600 BCE that rubbing amber (" electron" in Greek) with silk made the amber attract small objects like feathers. 
William Gilbert (1540  1603) found that many other substances produced similar effects when rubbed with a suitable material. The substances that behaved like amber were called "electrics".
Benjamin Franklin
(1706  1790) developed a theory of electricity which he modelled as a kind of fluid that flowed from one substance to another.
Franklin said, when electric fluid flowed from one substance to a second,
• the second substance was loaded (or "
charged") with an excess (+) of electric fluid, (positively charged) and
• the first substance was left with a deficiency () of electric fluid, (negatively charged).
Charles Du Fay (~1736) found that to explain both attraction and repulsion two kinds of electricity were needed. He called them vitreous and resinous. He observed that "like" kinds of electricity repel and "unlike" attract. The girl in the picture has the hairs on her head each with like charge and so they repel each other. 
Because two types of electricity were needed, Franklin's fluid theory was dropped, but the use of "positive charge" and "negative charge" were retained to distinguish the two kinds of electricity . These two kinds of electricity are present in equal quantities in neutral substances.
Auguste Coulomb measured the sizes of forces between charges with the apparatus shown on the right.

Coulomb’s law states that: The size of the electric force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of their separation and acts along the line joining them.
In symbols, where F is the electric force, q is the electric charge and r is the separation between the charges, this is: 

To get the units right in the S.I. system, for stationary charges in air or vacuum: 
• F is in Newton, 
Electric Potential Difference
Rocks fall downhill by gravitational potential energy producing motion, i.e. changing into kinetic energy.
Electric charge also falls downhill, but it will not be a gravitational slope but an electric one.
To review the idea of potential energy click here (for a page on potential energy from Peter's Physics Bridging course).
An electrical potential energy difference in a system =
the negative of the work done by the electric force.
An electrical potential difference in a system =
the negative of the work done per unit charge by the electric force.
Note carefully the difference in these two statements.
In symbols: The S.I. unit for potential difference is Joule/Coulomb. This is given the special name Volt (after Volta) 
Electric current
Electric current is due to an electric potential difference causing charges to fall down an electrical slope and change potential energy into kinetic energy.
Current is the flow of positive (or equivalent positive) charges. It measures how quickly charge moves through an area. 

The S.I. unit for current is, ,
which is given the special name, Ampere (symbol A).

Microscopic picture of current in a wire
To get an idea of the factors involved, imagine a multilane highway with cars moving at constant speed and no lane changes.
Let there be 2 people/car, let the density of traffic be 30 cars/km, and let each car move at a constant speed of 80 km/h.
For one lane at the end of one hour, a car which was 80 km down the highway will just be passing the given point.
All the cars stretching back that 80 km will pass during the hour and there will be 30 cars in each km.
Thus the number of people that pass a given point will be (2 people/car)×(80 km/h)×(30 cars/km) = 4,800 people/h.
With 3 lanes, this will be 14,400 people/h
A potential difference, V, across the ends of a wire, causes the free charges within the wire to accelerate.
However they soon hit a neighbouring atom and stop.
They are then accelerated again.
This start/stop motion results in the charges moving with an average small constant velocity, called the
drift velocity (v_{d}).
v_{d} corresponds to the speed of the cars in the illustration above.
The traffic density will correspond to the number of charge carriers per volume (n).
The people will correspond to the charge on each carrier (q).
The crosssectional area will correspond to the number of lanes (A).
The current that flows in a wire depends on: 
Example I1
The charge on each electron is 1.6x10^{19} C. Copper has 8.47x10^{28} electrons/m^{3}.
A copper wire has a radius of 0.5 mm and the electrons drift velocity is 3.5x10^{5} m.s^{1}.
Find the current in the wire.
Answer
Conductors/Conductance and Resistors/Resistance
A conductor is a device with the property of conductance,
i.e. it allows a certain level of current to flow through it.
A resistor is a device with the property of resistance, i.e. it restricts the flow of current.
These are simply reciprocal ways of looking at current flow.
In terms of external (extrinsic) factors, when current flows though a wire the drift velocity will be:
directly proportional to the potential difference between the ends (the height of the electrical hill), and
inversely proportional to its length. (The length determines how steeply the electrical hill slopes).
The drift velocity and external factors: 
Linking internal and external factors 
The proportionality factor (σ) will depend on the internal properties of the material.
G is called the conductance of the wire (in Siemens with symbol S)
σ is called the conductivity of the material (in Siemens.m^{1} or S.m^{1})
The conductivity is an intrinsic property because is determined only by the type of material.
The conductance of a wire is an extrinsic property, because its value depends on
both the shape and the type of material.
Note carefully the three different words.
Conductor  a device.
Conductivity  an intrinsic property depending only on the material.
Conductance  an extrinsic property depending on the size and shape.
Alternatively, a different proportionality constant (ρ) can be used.
where:
R is called the resistance of the wire (in Ohm with symbol Ω), and
ρ is called the resistivity of the material (Ohm.m or Ω.m).
Again, note carefully the three different words.
Resistor  a device.
Resistivity  an intrinsic property depending only on the material.
Resistance  an extrinsic property depending on the size and shape.
For metal wires, the relationship between current and potential difference is I = GV, or V = RI
Typical Resistivities at room temperature
Material 
Resistivity 
Temp. coef. of ρ (K^{1}) 
Conductors 

Silver 
16.2 nΩ.m 
+4.1 x 10^{3} 
Copper 
16.9 nΩ.m 
+4.3 x 10^{3} 
Aluminium 
27.5 nΩ.m 
+4.4 x 10^{3} 
Tungsten 
52.5 nΩ.m 
+4.5 x 10^{3} 
Iron 
96.8 nΩ.m 
+6.5 x 10^{3} 
Platinum 
106 nΩ.m 
+3.9 x 10^{3} 
Manganin 
482 nΩ.m 
+2.0 x 10^{6} 
Semiconductors 

ntype Si (Al) 
0.87 mΩ.m 

ptype Si (P) 
2.8 mΩ.m 

pure Silicon 
2.5 kΩ.m 
70 x 10^{3} 
Insulators or Dielectrics 

Glass 
0.01100 TΩ.m 

fused Quartz 
~10 PΩ.m 
Conductors, Insulators and Semiconductors
The table shows that Silver and Copper have a very low resistivity, which is why they are used in electrical equipment.
Platinum, although it looks like Silver is six times more resistant to current.
The third column records how temperature affects resistivity. Manganin is the least affected by temperature change.
This will be discussed in the next section.
As also seen from the table, resistivity separates materials into three main classes.
Conductors, that allow the free passage of charge through them.
Insulators, or dielectrics, that don't allow the free passage of charge though them.
Semiconductors, that are intermediate between conductors and insulators.
Ohmic conductors
A conductor is said to be an Ohmic conductor if its conductance and resistance are constant when the potential difference changes. 
Example I2
A tungsten filament at 20°C is 0.1 mm in diameter and 150 mm long.
Given that the resistivity of tungsten is 5.25×10^{8} Ω.m, find its resistance at 20°C.
Answer
The response of Resistivity to Temperature
Electrical resistivity changes with temperature. Some materials have a resistivity which increases with temperature (+ coefficients), others decrease ( coefficients). See the table above and graph below for details.
For metal wires, we can generally assume a linear increase with temperature, from room temperature to about 100°C. Carbon has a negative dependence as it's resistivity decreases with temperature. Other materials are fairly constant up to a specific temperature then change rapidly, so they can be used as temperature switches or thermistors.
Temperature coefficient of resistivity
For metal wires in the normal range 20° to 100°, the resistivity increases linearly as shown in the graph above.
The general equation for such a straight line is given by ρ = mT + b, where "m" is the slope and "b" is the intercept on the resistivity axis.
Usually, the resistivity equation is written with the axis at 20° and the resistivity there taken as a common factor:
here α (alpha) is the (linear) temperature coefficient of resistivity whose unit is 
Rewriting this: 
Rewriting again: This says that the fractional change in resistivity is proportional to the temperature difference, and α is the proportionality coefficient. 
Example I3
A tungsten filament at 20°C is 50 mm in diameter and 200 mm long.
Given the resistivity of tungsten is 5.25×10^{8} Ωm at 20°C,
and its temperature coefficient of resistivity is 4.5×10^{3} K^{1},
find the resistance at 2000K. (Assume any physical change in size can be neglected).
From above: 
This gives: 
Hence:
Power dissipated by a resistor
The tungsten filament in the previous question changed its temperature because of the heating effect of the current passing through it. Normally resistors are designed so that they dissipate heat at the same rate as it is produced so that their temperature remains the same. Assuming the resistance stays constant, the following shows the amount of power that a resistor will dissipate as heat.
Now (at constant temperature) V=RI so 
Example I4
An iron wire is 1.7 mm in diameter and 3 m long. It carries a current of 5 A.
Given that the resistivity, ρ, is 9.68×10^{8} Ωm, find the power dissipated by the wire.
Answer I4
Summarising:
There are two kinds of electric charge, positive and negative.
Electric charge is measured in Coulomb.
The smallest naturally occuring charge is 160 zC.
Electric current is the flow of electric charge down an electric potential gradient, and measured in Ampere.
Electric Potential Difference is the work done/charge in moving charges about, and measured in Volt.
The microscopic picture gives current depending on the charge density in the material, the size of the charge,
the speed that charges move through the material, and the area they move through., i.e.
A Conductor is a device that allows current flow, Conductance, G, and Conductivity, σ, are defined by:
A Resistor is a device that limits current flow, Resistance, R, and Resistivity, ρ, are defined by:
Conductors allow the free passage of charge through them.
Insulators (Dielectrics) don't allow the free passage of charge though them.
Semiconductors are intermediate between conductors and insulators.
Resistivity varies with temperature, the linear temperature coefficient of resistivity is given by:
Current flow produces heating. The power, in Watts, dissipated (as heat) by a resistor is given by:
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