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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.

Thales Postage Stamp 1994

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.

repulsion of hair

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.

In his honour, electric charge has the unit of Coulomb, (symbol C).

Today we know that the smallest charges that are found naturally are atom parts called "protons" (positive) and "electrons" (negative).

Protons and electrons have an electric charge of 1.6x10-19 C (or 160 zC) in the International System (S.I.)

portrait of Coulomb
Coulomb's balance

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:

Coulomb's law with proportionality constant

F is in Newton,
q is in Coulomb
r is in metre    

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:

electric potential difference

The S.I. unit for potential difference is Joule/Coulomb. This is given the special name Volt (after Volta)

Volta's  portrait

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.

definition of current

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

Andre Ampere was famous for finding the connection between electricity and magnetism.

portrait of Ampere

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 (vd).
vd 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 cross-sectional area will correspond to the number of lanes (A).

The current that flows in a wire depends on:
• the density of the charge carriers (the material).
• how much charge each carrier holds (the material).
• how quickly the carriers are moving (the potential difference).
• the cross-sectional area of the wire (the construction).

The first two are internal (intrinsic) factors and the second two are external (extrinsic) factors.

microscopic picture of current

Example I1
The charge on each electron is 1.6x10-19 C. Copper has 8.47x1028 electrons/m3. 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.


solution for current

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:

drift velocity

Linking internal and external factors

current and potential relationship

The proportionality factor (σ) will depend on the internal properties of the material.

separating internal and external factors

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.

current in terms of resistance


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



Temp. coef. of ρ (K-1)



16.2 nΩ.m

+4.1 x 10-3


16.9 nΩ.m

+4.3 x 10-3


27.5 nΩ.m

+4.4 x 10-3


52.5 nΩ.m

+4.5 x 10-3


96.8 nΩ.m

+6.5 x 10-3


106 nΩ.m

+3.9 x 10-3


482 nΩ.m

+2.0 x 10-6


n-type Si (Al)

0.87 mΩ.m


p-type Si (P)

2.8 mΩ.m


pure Silicon

2.5 kΩ.m

-70 x 10-3

Insulators or Dielectrics


0.01-100 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.
As shown in the graph, Current (I) ∝ Potential Difference (V)
This is usually written as "Ohm's law": V = RI,
where R is the Resistance.
Not all conductors follow Ohm's law.

graph of Ohmic conductor

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.


solution for resistance

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.

resistivity and temperature effects

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:

equation for resistivity

here α (alpha) is the (linear) temperature coefficient of resistivity whose unit is definition of temperature coefficient

Re-writing this:

re-written equation 1

Re-writing again:

re-written equation 2

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:

equation re-written


resistance solution

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.

power dissipated by a resistor

Now (at constant temperature) V=RI so

power dissipated in a resistor

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

solution for power


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.

current equation

A Conductor is a device that allows current flow, Conductance, G, and Conductivity, σ, are defined by:

conductance and conductivity definitions

A Resistor is a device that limits current flow, Resistance, R, and Resistivity, ρ, are defined by:

resistance and resistivity definition

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:

temperature coefficient of resistivity equation

Current flow produces heating. The power, in Watts, dissipated (as heat) by a resistor is given by:

power dissipated in resistor equation

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