Ohm's law, V-I graphs and ohmic/non-ohmic conductors

Plug a resistor into a circuit, vary the current, and plot voltage against current — the shape of that line tells you almost everything about how the material conducts. For some conductors the plot is a perfectly straight line through the origin; for others it bends, flattens, or even refuses to pass through the origin at all. Ohm's law is the name for the straight-line case, and it is not a universal law of nature but an experimental rule that some conductors obey and others don't. Learning to read a V-I graph — and to tell an ohmic conductor from a non-ohmic one like a filament lamp or a diode — is the core skill this article builds, alongside the microscopic reason a straight line appears at all.

The core insight

A conductor is ohmic only if its V-I graph is a straight line through the origin with constant slope $R$ , which happens exactly when the material's conductivity stays fixed as current and temperature change.

The core idea

Ohm's law states that for a conductor held at constant physical conditions — chiefly constant temperature — the potential difference $V$ across it is directly proportional to the current $I$ through it:

V \propto I \quad\Rightarrow\quad V = IR \tag{1}

where: $V$ is the potential difference (volts), $I$ is the current (amperes), and $R$ is the resistance (ohms, $\Omega$ ), the constant of proportionality — constant only if the conductor is genuinely ohmic. This is not just an empirical curve-fit. It follows from the microscopic picture of conduction, where free electrons drift under an applied field. The local (point) form of Ohm's law is

\vec J = \sigma \vec E \tag{2}

where: $\vec J$ is the current density (current per unit cross-sectional area), $\vec E$ is the electric field inside the conductor, and $\sigma$ is the conductivity of the material, a property that depends on temperature but not on $E$ or $J$ themselves for an ohmic material.

How the graph is read. In the standard verification circuit — battery, rheostat, ammeter, voltmeter, and the test conductor in series — the rheostat is adjusted to collect several $(I, V)$ pairs, and $V$ is plotted against $I$ . A straight line through the origin means the conductor is ohmic, and the line's slope is $R$ .

Ohmic vs non-ohmic. Metals such as copper, manganin, and nichrome are ohmic near room temperature: $\sigma$ stays essentially fixed, so $R = V/I$ stays fixed and the graph is straight. Several important devices are not:

A filament lamp heats up as current rises, and resistivity increases with temperature, so the curve bends and flattens — $R$ grows with $I$ .
A p-n junction diode conducts easily in forward bias and barely at all in reverse bias, giving a strongly asymmetric, non-linear curve that does not pass through the origin as a straight line.
Electrolytes and some other conducting media can likewise show non-linear or asymmetric $V$ - $I$ behaviour.

Because the ratio $V/I$ can change from point to point on a curved graph, it is worth separating two ideas: the static resistance $R = V/I$ at a particular operating point, and the dynamic resistance $dV/dI$ , the slope of the tangent to the curve at that point. For an ohmic conductor the two are identical everywhere, because the graph is one straight line; for a non-ohmic device they generally differ and both must be quoted with the operating point they refer to.

Figure

V-I graphs comparing a straight ohmic line through the origin with a bending non-ohmic filament curve and an asymmetric diode curve

Derivation AOhm's law from the microscopic current-density relation

The goal is to derive the familiar circuit-level form $V = IR$ starting from the microscopic relation $\vec J = \sigma \vec E$ , showing precisely why $R$ is a fixed geometric-and-material quantity — not an accident of the numbers — for a conductor with uniform cross-section held at constant temperature, and why that fixed $R$ is exactly what produces a straight V-I line through the origin.

Step 1. Write the microscopic law for a uniform conductor of length $L$ and cross-sectional area $A$ carrying current $I$ . The current density is $J = I/A$ (current spread uniformly over the cross-section), and since the field is uniform along a wire of length $L$ carrying potential difference $V$ end to end, $E = V/L$ .

Step 2. Substitute both into $J = \sigma E$ : $\dfrac{I}{A} = \sigma\,\dfrac{V}{L}$ .

Step 3. Rearrange for $V$ : $V = I\left(\dfrac{L}{\sigma A}\right)$ . Comparing with $V = IR$ , the resistance is identified as $R = \dfrac{L}{\sigma A} = \rho\dfrac{L}{A}$ , where $\rho = 1/\sigma$ is the resistivity.

Step 4. Read off the condition for ohmic behaviour. $R$ depends only on geometry ( $L$ , $A$ ) and on the material property $\sigma$ (equivalently $\rho$ ) — not on $V$ or $I$ themselves — provided $\sigma$ does not change as current flows. For metals at roughly constant temperature this holds, so $V$ and $I$ stay strictly proportional and the graph is a straight line through the origin with slope $R$ . The moment $\sigma$ starts to depend on $I$ (because self-heating changes the temperature, as in a filament) or on the direction of $E$ (as in a diode's asymmetric junction), $R$ is no longer a fixed number and the straight line breaks down — this is precisely the non-ohmic case.

Worked example

A torch bulb filament is tested and gives the readings $I = 0.1\ \text{A}, V = 1.0\ \text{V}$ and $I = 0.3\ \text{A}, V = 4.5\ \text{V}$ . Find the static resistance at each current, and compare it with the dynamic resistance between the two points to confirm the filament is non-ohmic.

Step 1. Static resistance at the lower current: $R_1 = V/I = 1.0/0.1 = 10\ \Omega$ .

Step 2. Static resistance at the higher current: $R_3 = V/I = 4.5/0.3 = 15\ \Omega$ . Since $R_1 \neq R_3$ , the ratio $V/I$ is not constant, so the bulb cannot be ohmic — its V-I graph is not a straight line through the origin.

Step 3. Estimate the dynamic resistance over this interval as $\Delta V/\Delta I$ : $\dfrac{\Delta V}{\Delta I} = \dfrac{4.5 - 1.0}{0.3 - 0.1} = \dfrac{3.5}{0.2} = 17.5\ \Omega$ .

Answer: static resistance rises from $10\ \Omega$ to $15\ \Omega$ as current increases, while the dynamic resistance over that stretch, $17.5\ \Omega$ , is even higher than either static value — exactly what is expected when a curve is bending upward, since the tangent slope outruns the chord from the origin. The physical cause is the filament heating up: rising temperature raises the metal's resistivity $\rho$ , so by Step 3 of the derivation above, $R = \rho L/A$ increases with current.

Competitive stretch

If a nichrome wire instead gives constant $V/I = 5.0\ \Omega$ across several readings (as in the standard verification experiment), the static and dynamic resistances coincide at $5.0\ \Omega$ everywhere — the signature confirmation of a genuinely ohmic conductor, and the case the derivation above describes when $\sigma$ truly stays fixed.

Common traps

Treating $V = IR$ as Ohm's law itself. $V = IR$ is just the definition of resistance and always holds, even for a diode. Ohm's law is the stronger physical claim that $R$ stays constant as $V$ and $I$ vary at fixed temperature.
Assuming every conductor is ohmic. Filament lamps, diodes, thermistors, and electrolytic cells are common devices that must not be forced into $V \propto I$ reasoning.
Confusing static and dynamic resistance. $V/I$ at a point and the local slope $dV/dI$ generally differ on a curved graph; they are equal everywhere only for a straight line through the origin.
Forgetting the constant-temperature clause. A wire that is ohmic at room temperature can turn non-ohmic if a large current makes it self-heat significantly, since $\sigma$ then changes with $I$ .

Ask the tutor: Want to work through more V-I data sets or sketch the diode and filament curves yourself with guided feedback?

Why isn't V = IR always the same thing as Ohm's law?

Because $V = IR$ is only a definition: whatever $V$ and $I$ happen to be for any device, dividing gives some number we call $R$ at that operating point. That is true for a diode, a filament, or a resistor alike. Ohm's law makes a stronger, testable claim on top of that definition — that this ratio $R$ stays the same fixed number no matter what $V$ or $I$ you choose, as long as temperature is constant. A device only "obeys Ohm's law" if it passes that stronger test, which is why filament lamps and diodes are called non-ohmic even though $V = IR$ still technically describes their static resistance at each point.

Why does a diode's V-I graph not even pass through the origin as a straight line?

A diode's conduction depends on the direction of the applied voltage in a fundamentally asymmetric way: in forward bias it conducts easily once past a small threshold voltage, and in reverse bias it conducts almost nothing across a wide voltage range. This means its conductivity $\sigma$ effectively depends on the sign and size of the field itself, not just on the material and temperature — so the microscopic condition behind Step 4 of the derivation (constant $\sigma$ ) fails badly and by design. The curve does still start at the origin at zero volts and zero current, but the extreme asymmetry and the near-flat reverse branch mean it is nowhere close to a single straight line.

If R = V/I always works as a number, why bother distinguishing static from dynamic resistance at all?

Because the two numbers answer different practical questions. Static resistance, $R = V/I$ , tells you the total opposition the device presents at that exact operating point — useful for power calculations like $P = VI$ or $I^2R$ at that point. Dynamic resistance, $dV/dI$ , tells you how much the voltage would change for a small extra nudge of current right there — useful when analysing small signal changes around an operating point, such as in diode circuits. For an ohmic conductor the distinction is moot since both equal the same constant $R$ , but for a curved graph like a filament or diode, quoting only one number without saying which kind can give a materially wrong answer to the question actually being asked.

Ohm's law, V-I graphs and ohmic/non-ohmic conductors

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads