V-I characteristics: linear and non-linear
An ohmic conductor gives a straight-line V-I graph through the origin. Diodes, GaAs and very hot wires give non-linear graphs, so Ohm's law has limits.
Linear and non-linear graphs
A V-I characteristic is a graph of the current through a device against the voltage across it. For a metal wire at a steady temperature, the graph is a straight line through the origin. Such a conductor is ohmic, and its resistance is the same at every voltage. Many devices are non-ohmic: their graph is curved, or different for positive and negative voltages. For them the ratio V/I changes with V, so a single value of R cannot describe them. Ohm's law is therefore not a universal law. It is a useful rule that many materials obey under steady conditions.
Three limitations of Ohm's law
| Limitation | What the graph shows | Example |
|---|---|---|
| V stops being proportional to I | Straight at first, then bends towards the V-axis at large currents | A metal wire or bulb filament heating up |
| The V-I relation depends on the sign of V | A large current for +V, almost none for −V of the same size | p-n junction diode |
| More than one V for the same I | The current rises, falls, then rises again as V increases | Gallium arsenide (GaAs) |
Ohmic vs non-ohmic
In a graph question, check two things: is the line straight, and does it pass through the origin? Both yes means ohmic.
How to draw the V-I graphs
- Draw V on the x-axis and I on the y-axis, crossing at the origin. Write the units (V, and A or mA).
- Ohmic: a straight line through the origin, into the first and third quadrants. Label it 'slope = 1/R'.
- Metal at large currents: straight near the origin, then curving towards the V-axis. Label it 'heating'.
- Diode: almost flat along the V-axis for small forward V, then rising steeply. On the negative side, a tiny current close to the axis. Label 'forward' and 'reverse'.
- GaAs: a curve that rises, dips, then rises again. Draw a dashed horizontal line cutting it at more than one point, to show one I at more than one V.
Worked example: is it ohmic?
(a) A wire gives these readings: V = 1.0, 2.0, 3.0 and 4.0 V; I = 0.20, 0.40, 0.60 and 0.70 A. Find V/I for each and comment. (b) A diode passes 20 mA at +0.80 V and 1.0 μA at −5.0 V. Find V/I in each case.
- (a) V/I = 1.0 ÷ 0.20 = 5.0 Ω; 2.0 ÷ 0.40 = 5.0 Ω; 3.0 ÷ 0.60 = 5.0 Ω.
- At 4.0 V: V/I = 4.0 ÷ 0.70 ≈ 5.7 Ω.
- Comment: R is constant up to 3.0 V, so the wire is ohmic there. At 4.0 V, R has risen, most likely because the wire has heated up.
- (b) Forward: V/I = 0.80 V ÷ (20 × 10⁻³ A) = 40 Ω.
- Reverse: V/I = 5.0 V ÷ (1.0 × 10⁻⁶ A) = 5.0 × 10⁶ Ω = 5.0 MΩ.
- V/I changes hugely with the sign of V, so the diode is non-ohmic.
(a) 5.0 Ω up to 3.0 V, then about 5.7 Ω: ohmic only at small currents. (b) 40 Ω forward and 5.0 MΩ reverse: non-ohmic.
Phone chargers and LED bulbs in Indian homes use diodes, which are non-ohmic on purpose: they let current pass one way only. An old filament bulb draws a large current just at switch-on, while its filament is still cold and its resistance is low.
2 marks: “State two limitations of Ohm's law. Give an example of each.” (a) V is not proportional to I at large currents, as in a heated metal wire. (b) The V-I relation depends on the sign of V, as in a p-n junction diode. (Or: more than one V for the same I, as in GaAs.)
1 marks: “What does the slope of the V-I graph of an ohmic conductor give, when V is on the y-axis?” Its resistance R.
2 marks: “V-I graphs (V on the y-axis) of two wires A and B of the same metal and length are straight lines through the origin. A's line is steeper. Which wire is thicker? Explain.” Slope = R, so R_A > R_B. With the same metal and length, R = ρl/A, so the wire with the smaller R has the larger area. B is thicker.
Straight and through zero means ohmic. Bends with heat, depends on the sign, or two V for one I means non-ohmic (hot wire, diode, GaAs).