Ohm’s Law Statement and Explanation
George Simon Ohm (1787-1854), a German physicist conducted a series of experiments to investigate the relation between the current passing through a conductor and potential difference between its ends. As a result of these investigations he arrived at an important conclusion which is known as Ohm's law after his name and may be stated as follows:
The current flowing through a conductor at constant temperature is directly proportional to the pd between its ends.
If I is the electric current flowing through a conductor maintained at a pd V, according to Ohm's law,
I ∝ V or V ∝ I;
i.e., V = IR;
where R is a constant known as the resistance of the conductor.
The graph connecting I and V is a straight line passing through the origin. If the temperature changes, then Ohm's law does not hold good because the resistance R of a conductor varies with change of temperature.
Unit of resistance: ohm (Ω)
Conductance (k) : It is the reciprocal of the resistance (R) of a conductor. k = 1/R
Unit: mho (Ω-1) or siemen (S).
Factors on which the resistance of a conductor depends
The resistance R of a conductor depends on:-
(i) the length l of the conductor R ∝ I
(ii) the area of cross-section a of the conductor R ∝1/a
(iii) the temperature t of the conductor (The resistance of a conductor increases with rise of temperature. But for semi-conductors like Ge, Si, etc. the resistance decreases with rise of temperature).
(iv) the material of the conductor.
Resistivity (ρ)
For a given material and at a given temperature, the resistance R of a conductor is
(i) directly proportional to the length l of the conductor i.e., R ∝ l ---------- (i)
and (ii) inversely proportional to the area of cross-section a of the conductor.
i.e., R ∝ l/a ------------ (ii)
From equations (i) and (ii),
R ∝ l/a
Therefore R = ρ(l/a)
Where, ρ is a constant called resistivity or specific resistance of the material of the conductor.
When l = 1 and a = 1, then ρ = R
Hence, the resistivity of the material of a conductor is defined as the resistance of the conductor of unit length and unit area of cross-section.
Unit of resistivity : ρ = Ra/l = ohm metre (Ωm)
Conductivity (σ)
Electrical conductivity is the reciprocal of the electrical resistivity.
σ =1/ρ
Unit of conductivity, σ = 1/ρ = mho/metre (Ω-1m-1) or Sm-1
Deduction of Ohm's law
Consider a conductor of length l, and cross-sectional area A containing n free electrons per unit volume. Let a pd V be applied between its ends.
The electric field along the wire E = V/l ---------- (i)
Acceleration of free electrons a= Ee/m --------- (ii)
where e = charge and m = mass of the electron.
If r is the average time interval between successive collisions of electrons with the ions of the conductor (relaxation time) the velocity acquired by the electron in time τ,
v = aτ = (Ee/m)τ, the drift velocity
The current through the conductor I = nAve = nA(Ee/m)τe
I = ne2Aτ/m x V/l (since, E = V/l) ---------- (iii)
Therefore, I ∝ V (since, ne2Aτ/ml = a constant)
Resistivity (ρ), Conductivity (σ) and Relaxation time (τ)
(i) From eqn. (iii), I = (ne2Aτ/m)(V/l)
Therefore, the resistance of the conductor, R = V/I = m/ne2τ x l/A
(ii) The resistivity of the material, ρ = m/ne2τ
(iii) Conductivity of the material, σ = 1/ρ = (ne2/m)τ
FAILURE OF OHM'S LAW
Ohm's law is not a basic law of nature. It simply represents the electrical behaviour exhibited by many materials. A true mathematical statement of Ohm's law is
V ∝ I
which means that a graph connecting V and I is a straight line passing through the origin. This relation holds good for metallic V conductors. Hence they are called Ohmic conductors.
There are many conductors in which the relation between V and I is different from the one given by Ohm's law. The conductors which does not obey the Ohm's law are known as non-Ohmic conductors. For these substance I — V graph is not a straight line passing through the origin. Circuits containing non-ohmic conductors are called non-ohmic or non-linear circuits.
In all non-ohmic circuits, there are some circuit elements having one or more of the following properties:
(a) Non-linear relationship between V and I
(b) The relation between V and I depends on sign of V for same magnitude of V.
(c) For the same current I, there may be more than one value of voltage V.
A few examples showing the violation of Ohm's law are given below:
(i) I — V graph for an Ohmic conductor is linear only for small currents. Even if the temperature of the conductor is kept constant, some conductors show an increase in resistivity as current increases.
(ii) A pn junction diode does not obey Ohm's law.
(iii) A thyristor or silicon controlled rectifier (SCR) does not obey Ohm's law. The SCR is a three terminal semiconductor switching device. When a pn junction is added to a junction transistor, we get a thyristor.
(iv) The flow of current through a gas in a discharge tube does not obey Ohm's law. For lower voltages, i.e., along OA Ohm's law is obeyed.
(v) The Ohm's law is not obeyed when current flows through an electrolyte if electrodes are of metals different from the one corresponding to the cations of the electrolyte.