Introduction :

Ohm’s law is the basic law regarding the flow of current. It was discovered by G.S. Ohm in 1828, long before the physical mechanism responsible for the flow of current - ‘electrons

The electrons were discovered by J.J. Thomson in 1897.

Ohm’s Law

Imagine a conductor through which a current is flowing

Let the V be the potential drop across the conductor. Then Ohm's  Law states that V α I

The potential drop across the conductor is directly proportional to the current. 

To remove the sign of proportionality we need to put some constant so. V= R I

Here. R= resistance of the conductor.

S.I. unit of resistance = ohm Ω

When current flows through the conductor it feels a resistance to its flow. The resistance of the conductor is independent of voltage or current.

There cannot be a better explanation of ohm’s law than the cartoon given below in the figure.

The figure describes how voltage is trying hard to push the current. And the resistance is trying to constrain the path of the current. We can draw two conclusions from it.

  • More the potential difference we apply on the ends of the conductor, the more will be the current passing through that conductor. V α I
  • More the resistance of the conductor, the lesser will be the value of current passing through the conductor. Current is inversely proportional to resistance.

Similarly, you can conclude that V is that factor that supports the current and R is that factor that opposed it                   

Resistance

The resistance of the conductor depends on

  • Length of the conductor
  • Area of cross-section
  • Nature of material of the conductor.

Now let's try to understand what caused the conductor to offer resistance to the flow of current and why it depends on the factors above. Let's try to get these answers first.

What causes the conductor to offer resistance to the flow of current?

In a conductor there are large numbers of electrons and ions, free electrons moving in the conductor suffer many collisions with the bound electrons, ions and other free electrons. These frequent collisions hinder the flow of electrons and thus provide resistance to the flow of charges and also provide resistance to the flow of electric current.

Why does resistance of conductor depend on the factors mentioned above?

I want to answer this with the help of an analogy. Imagine a very busy road where collisions with other vehicles are very frequent ( just a hypothetical situation ) with hundreds of vehicles on it. The longer that road is, the chance of more collisions. The same analogy is with a conductor, longer the conductor, more will be the number of collisions it would suffer and hence will offer greater resistance to the flow of current.   

Resistance of the conductor is directly proportional to the length of the conductor  R  α L

Now imagine two roads, one is a narrow road and the other will be wider. Suppose the same number of vehicles are traveling on both roads. Now tell me on which road the collisions between vehicles should be less narrow or wider? Obviously, the answer would be that collisions will be lesser on wider roads. You can answer it based on your daily experiences. In the same way, a conductor with a wider cross-section offers less frequent collisions than a conductor with a narrow cross-section

Hence resistance is inversely proportional to the cross-section area.

R α 1/A

The third dependence of resistance is on the nature of the material of the conductor.  Resistance depends on the charge density of the conductor and also some other factors like mean free path, relaxation time, and mass of the constituent atoms or molecules. These parameters are different for different metals. We account for all these factors in a single quantity called resistivity ρ of the conductor.

Therefore resistance is directly proportional to the resistivity of the conductor. 

R α ρ

If we combine all three we will get.

Resistivity:  Resistivity or specific resistance of a material may be defined as the resistance of a conductor of that material having a unit length and unit cross-section area. It is denoted by ρ, S.I. unit of resistivity is ohm meter  (Ω m). Resistivity depends on the nature of the material of the conductor and the physical conditions like temperature and pressure but it is independent of the shape and size

I- V characteristic and the resistance

I-V characteristic and the resistance are closely related, we can find the resistance of a conductor by using its I-V characteristic.

The slope of the I-V characteristic gives  1/R.

V= I*R     ;   I = (1/R) * V    

If we plot current along the y-axis and voltage on X-axis then, Comparing this with the equation of straight line y = m x, we can conclude that the slope of the I-V graph gives ‘ slope  m=1/R’, we can find resistance  R= 1/slope. refer to the graph above. The greater the slope of the I-V characteristics lesser is the value of resistance.

Now if we have a V-I graph, voltage is represented along the y axis and current along the ‘x-axis. Then on comparing the V= R I with  y= m x, we can conclude that the slope of the V-I graph is equal to the resistance. Greater the slope of V-I characteristics, the greater the value of resistance.

Current density, conductance and conductivity:

Let us quickly go through these definitions

Current Density: The current density at any point inside a conductor is defined as the amount of charge flowing per second through a unit area held normal to the direction of the flow of charge at that point. It is a vector quantity. Current density j= current/ area perpendicular to the direction of the current.

J= I / A

If the area is not perpendicular to area A but makes an angle θ  with the direction of current then.

Component of Area normal to the current  ‘An’= A cosθ 

J= I/ Acosθ 

S.I. Unit = Ampere per square meter. ( A/m^2)

Conductance: The conductance of a conductor is the ease with which electric charges flow through it. It is equal to the reciprocal of resistance. It is denoted by  G

Conductance = 1 / resistance

G= 1/ R

S.I unit is  mho or siemens (S)

Conductivity:  The reciprocal resistivity of a material is called conductivity. It is denoted by σ.

Conductivity = 1 / resistivity

σ = 1/ρ

S.I. Unit is siemens per meter. (S/m)

Vector form of ohm’s law.

If E= magnitude of the electric field in a conductor

 l= length of the conductor, A= Cross-section area of conductor

V= potential difference, ρ= resistivity, σ= conductivity

R= Resistance, J= current density

Then

The above expression J=  σ E is called the vector form of ohm’s law.

Limitations of Ohm’s Law

  •  This law is not applicable to unilateral networks. Unilateral networks allow the current to flow in one direction. Such types of networks allow the current to flow in only one direction.

For example, transistors, diodes etc.

  • This law is not applicable for non-linear electrical elements with parameters like capacitance, resistance etc. voltage in such circuits won't be constant with respect.
  • Ohm’s law is only applicable for metallic conductors and not valid for non-metallic conductors.

I-V characteristic of various electrical elements.

Ohmic and Non-ohmic conductors

In ohmic conductors, the graph between Voltage and current is linear. The graph starts at the origin and is a straight line. These conductors follow ohm’s law. I-V characteristic graphs of non-ohmic conductors are non-linear graphs, usually curved ones.

Note: A straight-line graph with an intercept ( not starting from the origin ) is also a non-linear graph.

Drift velocity, mobility and relaxation time for the conductors

When we apply electric field ‘E’ across the end of the conductors. Free electrons experience a force on it  ‘F= -eE’ and thus also experience an acceleration. F= ma= -eE so a = -eE/m, This is the acceleration experienced by the charge in the presence of an electric field.

The electron in the conductor will start to drift in a direction opposite to the direction of the Electric field. But the electrons would still suffer multiple collisions but now there will be a preferred direction of motion of electrons. The average time between two successive collisions of an electron is called relaxation time 'τThis relaxation time is of the order of 10^(-14)s.

The velocity gained during this time is called drift velocity ‘Vd’ Vd= a* τ = -eEτ/m

Drift velocity may be defined as the average velocity gained by the free electrons of a conductor in the opposite direction of the externally applied electric field.

Mobility of charge carriers

The conductivity of any material is due to its mobile charge carriers. Greater the mobility of a charge carrier will be its conductivity.

The mobility of a charge carrier is the drift velocity acquired by it in a unit electric field. I

Mobility is denoted by μ, and  μ= Vd/E  = eτ/m

Expression of current in terms of drift velocity and mobility

We can express an electric current in terms of drift velocity and mobility also.

I= neA *Vd = neA* μ E

Where  I= electric current

n= free charge density of  the conductor

A= Area of a cross-section of conductor

Vd= drift velocity

μ= mobility of free electron

E= applied electric field.

Expression of resistivity in terms of electron density and relaxation time

We can get the resistivity in terms of electron density and relaxation time. We would start with the formula of drift velocity and current in terms of drift velocity. We will then rearrange the terms and use ohm’s law.

The above expression is the formula for resistivity

It depends on two factors :

  1.  Number of free electrons per unit volume or electron density of the conductor
  2. The relaxation time  τ

The resistivity of different materials.

The materials are classified as conductors, insulators and semiconductors.

Conductors: The materials which conduct electric currents are conductors. The resistivity of the conductors is very low. It ranges from 10^(-8) Ω m to 10^(-6) Ω m.

Copper and aluminum have the lowest resistivity. Nichrome has a resistivity of about 60 times that of copper so nichrome is used in electric heaters and electric iron. Metals have a positive coefficient of resistivity. With the increase in the temperature of metals their resistivity increases.

Insulators: The materials which do not conduct electric currents are insulators. They have high resistivity, more than 10^4 Ω m. Insulators like glass, mica, and bakelite have very high resistivity in the range of 10^(14) Ω m to 10^(16) Ω m. Semiconductors: These are the materials whose conductivity lies in between conductors and insulators. They usually conduct at room temperature and are insulators at very low temperatures. The range of resistivity of semiconductors is very wide and ranges from 10^(-6)  Ω m to 10^(4) Ω m. 

Semiconductors have a negative temperature coefficient of resistivity. The resistivity of semiconductors decreases with an increase in temperature.

Temperature dependence of the resistivity

 If we see the formula for resistivity we do not see temperature explicitly present there. Then why does the resistivity depend on temperature?

The answer to this question is that resistivity depends inversely on relaxation time. The relaxation time is the average time elapsed between two successive collisions. This relaxation time actually depends on temperature.

As the temperature increases, there will be more frequent collisions and thus reducing the relaxation time. If you follow the formula for resistivity, you can tell that resistivity is inversely proportional to relaxation time. If the relaxation time would decrease with increase in temperature. Resistivity would increase with increase in temperature.

So. Yes! There is no explicit dependence on temperature in the formula of resistivity, but it is implicitly present during the relaxation time.

In the above text, I have explained why the temperature affects the resistivity of a conductor. Now the next question will be How much does the temperature affect the resistivity of any conductor?

Resistance and resistivity both increase with the increase in temperature in exactly the same manner. As they are Resistance is directly proportional to resistivity. In the figure given below, there is given the formula.

If we consider the dependence of resistance on temperature

Here R= resistance at temperature T

R0= resistance at temperature To

To= reference temperature is usually 20 degrees Celcius but sometimes it is 0 C.

T= given temperature

α= Temperature coefficient of resistivity.

Temperature coefficient of resistivity

We can find the temperature coefficient of resistivity by just rearranging the terms. Consider,

Note: the value of α is positive for conductors and negative for semiconductors.

For conductors, the value of resistivity and hence resistance would increase with the increase in temperature. As shown in the figure below. At higher temperatures, the slope of the V-I graph will be more.

Electrical power and energy

To maintain the steady flow of current through the conductor in a circuit an external force is required which must supply the power. In a simple circuit with a cell, It is the chemical energy of the cell which supplies this power. Moreover, Inside the conductor when free charges are drifting inside the conductor under the action of the electric field, their kinetic energy would increase as they move, However, we have got that charges do not move with acceleration but move with steady drift velocity.

This is because of the collision of the ions and the atoms during transit. The kinetic energy gained by the charges is shared by the atoms during collisions and atoms start vibrating vigorously, and the conductor heats up. Thus an amount of energy is dissipated as heat in the conductor. The energy dissipated per unit time is called power dissipated. The formula for the power dissipated is given below.

Where R= resistance of the conductor, V= potential difference and I= current through the conductor. If you are someone who always struggles between energy and power. Let me try to help you with that.

Please look at the diagram above, It relates energy and power with water flowing through a pipe and collecting in the bucket.

Power is actually the rate of doing work or here it can be related to the rate of flow of water and energy is the amount of work done in some time ‘t’ and here it is related to the amount of water collected in the bucket in time ‘t’. We can conclude the discussion above about electrical energy and power in a table.

The above diagram can help you to remember all these formulas.

A fun thing to do: Below is the link to the simulation of ohm’s law, you can play with it.

simulations on ohm's law

Explanation of the simulation: In this simulation, there is a circuit with some battery and resistor and you can see the values of V, I and R.

What you do in this

  • You can change the value of voltage by sliding the voltage up and down can note how current is changing
  • You can change the value of resistance in the circuit for a fixed value of V and see how current will be varying.