Resistivity and Electrical Conductivity – The Complete Guide

Ultimate Guide on Resistivity and Electrical Conductivity

We know what the electric wires are made of; they are either made of copper or aluminum. We also know that Gold and Silver would have been a better choice, had they been cheaper. However, the question is what makes them suitable for the same? Why don’t we see lead wires? The answer lies in the very basic property of these materials, namely their resistivity and electrical conductivity. This article explains the essential concepts of resistivity and electrical conductivity. Resistivity and conductivity are actually two sides of the same coin, if you understand one, you will get the other as well.

What is Resistivity?

The resistivity of a material is the measure of its property due to which it opposes the flow of electrons through it. 

As we know, the flow of electrons leads to current flow, therefore if a material opposes the flow of electrons, the current that can pass through it is limited. Some materials oppose the flow more than others. This is due to the varied atomic structure of different materials.

In order to understand resistivity better, let us first revise the concept of ohm’s law.

OHMS LAW:

The ohm’s law states that when a voltage (a potential difference) is applied across a conductor, current starts to flow through it. This current is directly proportional to the voltage.

Numerically, Ohms law can be written as:

V α  I;  where  V = voltage, I= Current through the conductor

Or V= RI                     …….  equation  (1)

Here, R = Constant, known as Resistance.

This resistance restricts the amount of current or we can say that it restricts the amount of electron flow. This means that the conductor resists the current flow to some extent.

SI UNIT OF RESISTANCE

R = ohm denoted by Ω. = Volt/Amp.

In order to change the amount of current, that is the amount of electrons flowing, we can just vary the resistance value.  You may assume, from equation 1, that to increase the resistance, we just have to increase the voltage applied. Well, you are wrong!, as according to the ohms law, an increase in voltage applied will only result in an increase in current( voltage and current are proportional to each other, remember! And ‘resistance’ R is a constant for any given conductor of fixed length and area) and would give us no change in resistance. Just look at the figure below that shows voltage versus current graph. It is a straight line with a slope R.

Since now we know changing the applied voltage won’t help changing the resistance, then what can be done to change it? For that, let us look into the factors that affect the resistance of the conductor.

 Factors affecting Resistance:

The resistance of a material is its ability to oppose the flow of electrons through it. It depends on the physical dimensions; that is its Length and Area of the cross section. Other factors include temperature and type of material used to make the conductor.

First, let us analyse the effect of the length and area of cross-section on the resistance of a conductor.

For that, we can take example – a conductor of length L and area of cross-section A having a resistance R.

Case 1: Changing Length of Conductor (and Area of cross section kept constant).

From the electromagnetic theory, whenever a voltage, V is applied across a conductor, an electric field, E is formed. The two are related by the following equation:

 V=E.L         …..          equation (2)

L is the length of the conductor.

Now as we increase the length, L, what will happen to the electric field? From equation 2,  it is clear that it will decrease ( as the voltage applied, V is held constant).

If the Electric field decreases, the electrons would start to move in a haphazard way due to a weaker electric field.  It results in collision of electrons, only to disrupt the electron flow. Thus, the resistance towards the electron flow increases, that means the resistance of the material increases.

Hence we can say that an increase in length, would increase the resistance.

Similarly, when we decrease the length, L can you guess what will happen? Yes, the converse of above actions takes place. Lesser the length, more the electric field and there would be a better electron flow. This means with a decrease in length, the resistance of the material to the flow of electron decreases.

Hence a decrease in length would decrease the resistance.

From above two discussions, we can sum it up that the resistance is directly proportional to length, L.

A rheostat works on this principle. The effective resistance is changed by varying the effective length of the resistor. An image representing the same is given below.

Case 2: Changing the cross-sectional area A (and the length L is kept constant).

Here, since the length is kept constant, the electric field does not change (as the voltage is also constant).  Therefore, to understand how the area of cross section affects the resistance, we can use a simple analogy of a water pipe.

Think of the conductor as a cylindrical pipe and the electrons as the water.

Let the water flow through the pipe. Now as we increase its cross sectional area, we can see that the amount of water that flows through it has increased.  See the figure for reference.

Similarly, as we decrease the cross-sectional area, the amount of water that can flow through the pipe decreases.

Similarly, in a conductor, when its area of cross-section is increased, the flow of charges[current] increase, meaning the opposition to the  movement of charges decreases, hence there is a decrease in resistance.

Thus, increasing the area of cross-section decreases the resistance.

Also, when the area of cross section is decreased, the space available for the electrons to flow decreases, and therefore the opposition to the electron flow increases, meaning there is an increase in the resistance.

From the above discussion, it is pretty clear that resistance is inversely proportional to the area of cross section, A

It is for this reason that the resistance of a wire and that of a sheet is different, even though both are of same length and material. The same case is represented using an image below.

Resistivity Equation (Formula)

After analyzing the two cases, we know that the resistance is directly proportional to length and inversely proportional to A. Mathematically, it can be represented as

We have analyzed the effect of length and area of cross-section on resistance, but what about the other two factors, the temperature, and type of material? For this, proportionality constant is introduced in equation 5. This constant is called the resistivity. Since for a given material at a given temperature, the resistivity is constant, it is taken as the proportionality constant here. So finally let’s see the resistivity equation or a mathematical formula to define resistivity.

It is denoted by a Greek letter, rho, ρ. The resistance, R, now can be written in terms of the following

Let us now understand the relevance of resistivity.

The relevance of resistivity:

It is the measure of the ability of how much a material can oppose the flow of electrons.

Knowing the resistivity of a material, we can choose it accordingly, for different uses. An insulator such as a glass has a high resistivity, whereas that of a good conductor like copper has a very low resistivity, and hence is suitable for making connecting wires.

For a conductor with electric field E(produced by an applied voltage), that results in a current density of J inside it, the resistivity can be defined as follows:

SI UNIT of Resistivity, ρ 

In this section, let’s see more about resistivity units.

The SI unit of resistivity can be found from the equation (7),

We know SI UNIT of E = Volt/m   And  SI UNIT of J = Amp/m²

Then from equation 7,

SI UNIT OF ρ = SI UNIT OF E/ SI UNIT OF J

Thus from the equation, SI UNIT OF Resistivity, ρ  = Ωm

Even though the resistivity(ρ) is taken as a constant for a given material, but really is it? Is there anything that can change it? Yes, there is. Let us see what factor the resistivity depends on.

Factors affecting resistivity:

Since ρ is defined for a particular material, so yes the type of material is one factor that affects it. It is due to the fact that different materials have different atomic and molecular arrangements. If we had to increase the resistivity of that material we could just increase the amount of material used. But then it is not that common a practice, since it would make the conductor nothing but bulky.

Another factor that affects the resistivity is the temperature. Every material has a temperature coefficient, and resistivity is dependent on it. It is due to this reason, whenever the resistivity of a material is given, the temperature at which it was measured is specified. If the temperature is not specified, it is assumed to be at the room temperature.

A positive temperature coefficient implies that the resistivity increases with temperature. Whereas a negative coefficient implies that it decreases with temperature. Metals such as copper, have positive temperature coefficient whereas that of semiconductors like Silicon, have negative temperature coefficient.

A linear approximation, gives the following relation between resistivity, ρ(T) and temperature coefficient, alpha, α, at some temperature, T:

ρ₀ is the resistivity of the material at a reference temperature(mostly the room temperature), T₀.

Thus, the knowledge of the resistivity of the material is vital for making the correct choice of material to be used.

Nichrome, an alloy of nickel, chromium, and iron, has a resistivity ranging from 1.10 × 10⁻⁶ Ωm to 1.50 × 10⁻⁶ Ωm  and is best suited for making resistors.

Lead has a resistivity of 2.20×10⁻⁷  Ωm, and is a poor conductor of electricity and hence not used in electric wires.

Now as we have covered the basics of resistivity, let’s have a look at some practical examples before we proceed to know about electrical conductivity.

Measure Resistivity (Practical Examples)

Example No1:

You have to calculate the resistance of a rectangular strip of aluminium whose  dimensions are as follows: Length: 1.5m; Width:20mm; Thickness :0.5 mm; The resistivity of aluminium is given as 2.65×10^-7 Ωm(at 20⁰C).

Here, we will use the equation (6) that is :

We already know from the given values that ρ = 2.65×10^-7 Ωm and L, Length = 1.5m

We have to calculate the area of cross-section. Since we have a rectangular strip, the area of cross section is given by

A= width×thickness = (20×0.5)mm² = 10mm²= 10 ×10^-3×10^-3 m²=1×10^-5 m²

Now that we know all the values from the equation 6, we can now calculate the resistance, R

Example No2:

You have been given a coil of copper wire of length L and cross sectional area of A. The length of the coil is now reduced to half. Will the resistance of the wire change? If yes, then by what factor will it change?

The resistance of the wire is given by equation 6. From that, we can say that a change in length does change the resistance.However, by what factor it has changed, we have to find out:

Now, here we see that since the copper wire has not been changed to some other material, and assuming that the temperature has not been tampered, we can say that the resistivity, ρ remains constant. Since only the length has been reduced to half, the thickness has not changed. Therefore, the area of cross-section, A remains constant.

Let L be the original length. Then L/2 will be the new length of the coil. Let R be the original resistance and R^’ be the new resistance. Then

Dividing both we get

Thus by reducing the length by half, the resistance of the wire is reduced by half that isit is changed by a factor of ½.

Electrical Conductivity

The electrical conductivity of a material is the measure of the property of a material due to which it allows the electrons to flow through it.

From the definition, it is clear, that electrical conductivity is actually the converse of resistivity.  There is another parameter called conductance, in an electric circuit. It is the measure of how much the current can be conducted through the conductor. The electric current, as we know is the flow of electrons. In simple terms, it is the opposite of resistance. If resistance is the factor that restricts the current in an electric circuit, conductance is the factor that allows the current to flow.

Conductance is often represented as G = 1/R.

SI Unit Of Conductivity and Conductance

Since, G = 1/R.

SI Unit of G = 1/ SI unit of R

= 1/Ω

= Ʊ, mho[ known as seimens, often written as S]

Therefore

SI Unit of conductance =  Seimens, S

Now let’s see what affects the conductance of a material.

Factors affecting Conductivity and Conductance:

Since conductance is just the opposite of resistance, the factors that have an effect on conductance remain the same as that of resistance, but how they affect changes, it just becomes opposite.

1. Length , L: As length increases the conductance decreases.

G α 1/L

2. Area of the cross-section , A: As area of cross-section increases the conductance increases.

G α A

Thus: G α A/L

Conductivity Equation (Formula)

Here a proportionality constant, named as conductivity is introduced.

It is denoted by the symbol sigma, Ϭ and is the reciprocal of resistivity.

Ϭ = 1/ρ

Resistance, R can be written in terms of conductivity, Ϭ as:

From the equation, it is clear that if the conductivity of the material is high, the resistance is low and vice versa.

This knowledge of conductivity helps to determine the good and bad conductors of electricity. If the conductivity of a material is high enough, it is a good conductor of electricity, so it allows more charge to flow through it.

Also like resistivity, conductivity also depends on temperature and is specified at a particular temperature.

Metals like copper, aluminum, gold, and silver have high conductivity in the range of MS/m, making them good conductors of electricity and suitable for making electrical wires.

Now that we have discussed the concepts in detail, let us do a quick revision, but before that let us look into some practical examples related to conductivity.

Measure Conductivity (Practical Examples)

Example No3:

You have a copper wire of length 5 m and area of cross section 20mm². The resistance of the wire is given as 400mΩ. Calculate its conductance and conductivity.

As we know, conductance, G is the reciprocal of resistance R.

G= 1/R = 1/400mΩ =  2.5 S

Also from resistance and conductivity relation(equation 10):

Example No4:

You have been given a rectangular strip of length L and area of cross section A. The thickness of the strip is increased so that the area of cross-section is increased to 2A. What would happen to the conductance of the strip?

As the area of cross section is increased to 2A and length is same, from equation 9, it is clear that the conductance will also increase two-fold ,so that the new conductance will be 2G.

Resistivity and electrical conductivity- In a nutshell:

• Resistivity, ρ- Property of a material to “oppose” the flow of charge.
• Conductivity, Ϭ – Property of material to “allow” the flow of charge.
• Resistance, R –  directly proportional to length, inversely proportional to area of cross-section.
• Conductance, G – inversely proportional to length, directly proportional to area of cross-section.
• G = 1/R
• Ϭ= 1/ρ
• SI Unit of R = ohm, Ω
• SI Unit of G = Siemens, S
• SI Unit of ρ = Ωm
• SI Unit of Ϭ = Seimens per meter, S/m
• The resistivity of a material helps to classify them into conductors, semiconductors and insulators. For conductors the resistivity will be very low while that for an insulator it will be very high.
• The conductivity of a material helps to classify the conductors into good and bad conductors of electricity. The good conductors will have high value of conductivity.
• Both conductivity and resistivity are temperature dependent.

So that’s it! We have finished our guide on Electrical Resistivity and Conductivity. If you have doubts, ask in comments. If you liked our guide, please share with your friends.