Limitations of Ohm's Law

Current Electricity · CBSE

Theory

Ohm's law is not a fundamental law of nature. It came out of a model in which the relaxation time $\tau$ and the free-electron density $n$ are constants independent of the field; where that assumption fails, so does the law.
Ohm's law holds for many substances over a range of fields. If the field becomes too strong, departures appear in all cases — even in homogeneous conductors like silver and in pure or doped germanium.

(a) $V$ ceases to be proportional to $I$ . The $V$ – $I$ graph starts along the dashed straight line of Ohm's law near the origin and then bends away from it. This is what happens when $\rho$ rises with $I$ even at fixed temperature.
(b) The relation between $V$ and $I$ depends on the sign of $V$ . Reversing the polarity while keeping the magnitude of $V$ fixed does not give a current of the same magnitude in the opposite direction. A junction diode is the standard example, and its characteristic is drawn on different scales for the forward and reverse halves.
(c) The relation is not unique: more than one value of $V$ gives the same current $I$ . Gallium arsenide (GaAs) behaves this way.
A rectifier combines features (a) and (b); GaAs shows feature (c).
These non-ohmic materials and devices are not curiosities — they are what electronic circuits are built from.

Sketch $I$ on the vertical axis against $V$ on the horizontal axis. From the origin the curve rises: the first portion, labelled AB in CBSE's figure, is a straight line through the origin, so Ohm's law is obeyed there.
Beyond that the rise slows and becomes non-linear (region BC), and the curve reaches a peak around region CD.
After the peak the current falls as the voltage keeps increasing — region DE. Here $\mathrm{d}I/\mathrm{d}V < 0$ , so the resistance $\mathrm{d}V/\mathrm{d}I$ is negative. This is the region of negative resistance.
The same reading applies to a resistance-versus-voltage sketch labelled with other letters: whichever portion has the current falling while the voltage rises is the negative-resistance region.

Ohmic conductor	Non-ohmic device
$V$ – $I$ graph is a straight line through the origin	Graph is curved, or unsymmetrical, or many-valued
$R$ is the same at every point of the graph	$R$ changes from point to point on the graph
Reversing the polarity reverses the current with the same magnitude	Reversing the polarity may give a quite different current
Metallic conductors at constant temperature	Diode, transistor, thermistor, electrolyte, GaAs

Three deviations: non-linear $V$ – $I$ ; sign-dependent $V$ – $I$ (diode); non-unique $V$ – $I$ (GaAs).
Negative resistance is the region where $I$ decreases as $V$ increases.
Ohm's law holds in the linear region through the origin.
$V = IR$ still defines the resistance of a non-ohmic device at a given operating point; it is Ohm's law, not the definition, that fails.

Real board questions. Answers are at the end of the page.

1
The graph showing the variation of current versus voltage for a material GaAs is shown in the figure. Identify the region of (i) negative resistance (ii) where Ohm's law is obeyed.
1 markCBSE Class 12 Physics — Very Short Answer · Delhi 2015, Ap · Section A, 1 markAnswer 1 ↓

Work the questions above first, then check yourself here. Each answer is set out the way it should be written in the paper.

1.↑ back to the question
(i) In the region $DE$ of the graph, the current decreases as the voltage increases, i.e. $dI/dV < 0$ , giving a negative resistance. (ii) In the region $AB$ , the graph is a straight line through the origin, so Ohm's law ( $V \propto I$ ) is obeyed there.