Problem — reading Ohm's law from a graph

Problem

Reading Ohm’s law from a graph. An ohmic resistor was characterised by varying $V$ and measuring $i$:

The measured points (red) lie along a straight line through the origin (blue).

(a) Why must an ohmic conductor give a straight line? (b) How do you obtain $R$ from the graph? (c) If the curve bent downward for large $V$, what would you deduce about the conductor’s behaviour (e.g. a lamp filament)? Inspired by (O’Kuma et al. 2004).

Solution

(a) Why a straight line. A conductor is ohmic when its resistance $R$ is constant (it does not depend on $V$ or $i$). Ohm’s law $V = R\,i$ is then a linear relation between $i$ and $V$: in the $(V, i)$ plane it is a straight line through the origin (at zero voltage the current is zero). Linearity is the experimental signature of ohmicity.

(b) How to obtain $R$. Here $V$ is on the horizontal axis and $i$ on the vertical, so the slope of the line is $\text{slope} = \frac{\Delta i}{\Delta V} = \frac{1}{R}$ and the resistance is the inverse of the slope, i.e. $R = \Delta V/\Delta i$. From the graph the line rises by about $\Delta i = 2{,}75\ \text{A}$ over $\Delta V = 5{,}5\ \text{V}$: $R = \frac{\Delta V}{\Delta i} = \frac{5{,}5}{2{,}75} = 2{,}0\ \Omega$ (In general, if the line had $\Delta V/\Delta i = 12$, it would be $R = 12\ \Omega$: the recipe is always “resistance = inverse of the slope”.)

(c) Curve bending downward. If for large $V$ the current grows less than proportionally (curve concave downward), it means that for the same increase in $V$ the current increases less: the resistance increases. This is the behaviour of a lamp filament, which heats up through Joule heating and increases its resistivity: it is no longer ohmic and the characteristic curves.

$\ev{\text{(a) linearity } V=Ri\;;\quad \text{(b) } R = \Delta V/\Delta i \approx 2{,}0\ \Omega\;;\quad \text{(c) } R \text{ increases with heating}}$

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Topics: Circuiti elettrici Concepts: Legge di Ohm Skills: Lettura dei grafici