Ohmic & Non-Ohmic: I–V Characteristics

Curve reading is a daily professional act. Solar-cell engineers live on the I–V characteristic: a cell's maximum power point — the knee of its curve — is what inverter electronics chase from moment to moment as clouds pass, using exactly the static-versus-dynamic distinction of this module. Battery engineers read a cell's internal state from how its characteristic sags under load. And every electronic component you can buy ships with its characteristic curves printed in the datasheet — an LED's version of the diode knee sits near 2–3 V depending on colour, which is why every LED you have ever seen has a series resistor standing guard against the near-vertical cliff. The curve-tracer instrument — a machine that sweeps a component and draws its I–V signature on a screen — remains the standard tool for spotting counterfeit or damaged parts: a fake transistor confesses in one sweep, because the one thing a component cannot do is forge its own characteristic.

Imagine testing four unlabeled components with the rig from Ohm's Law & Resistance: sweep the voltage, log the current, plot. Four different objects, four unmistakable curves:

With current on the vertical axis — the convention this article and most modern papers use — a simple dictionary translates shapes into physics. Steep means willing: a large current for a small voltage, so low resistance. Flat means stubborn. So the filament's curve, which starts steep and flattens, is a component whose resistance grows as it operates — because the operating heats the tungsten from room temperature toward 2500 °C, and hot metal resists more (the collision story from Drift Velocity & the Microscopic Picture). The thermistor's curve, steepening as it rises, is the mirror image: it is a semiconductor, heat liberates more charge carriers than it obstructs, so warming it makes it conduct better. The diode is a different animal altogether — not a heater but a one-way gate with an opening price: essentially no conduction until the forward voltage reaches the knee (about $0.6$–$0.7\ \mathrm{V}$ for silicon), then an almost vertical rise; reversed, it passes only a leak measured in microamps. One reading protocol before anything else, because half of all dropped marks on this topic die here: read the axis labels first. Some books and boards (NCERT among them) draw $V$ on the vertical axis and $I$ on the horizontal. Every visual verdict then flips: the gradient of the ohmic line becomes $R$ rather than $1/R$, and the filament's curve bends steeper, not flatter. The physics has not changed — only the mirror it is viewed in. Decide which plot you are looking at, translate, then judge.

For the ohmic line, everything is inherited: $V = IR$ with constant $R$, from Ohm's Law & Resistance. Plotted as $I$ against $V$ it is

where: the bracketed slope $1/R$ is the line's gradient — the conductance, if you met that word. A $20\ \Omega$ resistor draws a line of gradient $0.05\ \mathrm{A/V}$; a $5\ \Omega$ resistor draws one four times steeper. Two lines on one grid rank themselves instantly: steeper line, smaller resistance — provided $I$ is vertical. This single sentence answers a whole family of MCQs. Now the tool that curved characteristics force on us. For any point on any characteristic, two different "resistances" exist, and JEE-Main tests whether you know they differ:

where: $R_{\text{static}}$ — the ratio of the coordinates of the operating point (graphically: linked to the line from the origin to the point); $r_{\text{dynamic}}$ — the ratio of small changes about that point (graphically: the local slope of the curve read as $\Delta V/\Delta I$, the reciprocal of the tangent's gradient on an $I$–$V$ plot). For an ohmic conductor the two are equal everywhere — that is just another way of saying the characteristic is straight through the origin. For anything curved they part company: a silicon diode passing $10\ \mathrm{mA}$ at $0.70\ \mathrm{V}$ has $R_{\text{static}} = 70\ \Omega$ but a dynamic resistance of only a few ohms, because near the knee a tiny extra voltage unleashes a large extra current. Static answers "what is the ratio here?"; dynamic answers "how does it respond to a nudge?" — the question that matters for signals and for rating small changes.

where: the subscripts run over the measured data points. This is the table form of "straight line through the origin," and it is how you certify or convict a component when the exam gives numbers instead of a picture — compute the ratio for every row and watch for drift. A constant ratio is ohmic; a trending ratio is a confession, and the direction of the trend names the culprit (rising = metal heating; falling = thermistor-style semiconductor).

The filament lamp. At switch-on the tungsten sits at room temperature with a small resistance; the first surge of current heats it toward operating temperature and the resistance climbs several-fold (the mechanism — hotter lattice, faster collisions — is Drift Velocity's picture). On the $I$–$V$ plot the curve therefore starts with the steep gradient of the cold lamp and flattens as heating throttles the current. Numbers make it vivid: a $230\ \mathrm{V},\ 100\ \mathrm{W}$ bulb runs at $R_{\text{hot}} = V^2/P = 529\ \Omega$, but its cold filament measures roughly a tenth of that, near $53\ \Omega$ — so the first instant after switch-on drives about $4.3\ \mathrm{A}$ through a filament rated for $0.43\ \mathrm{A}$. That ten-to-one inrush is why bulbs usually die at the flick of the switch, not mid-evening.

The junction diode. Forward biased, it conducts negligibly until the applied voltage overcomes the junction's built-in barrier — the knee voltage, about $0.6$–$0.7\ \mathrm{V}$ for silicon (germanium sits near $0.3\ \mathrm{V}$) — after which current climbs extremely steeply. Reverse biased, only a microamp-scale leakage flows. The characteristic is therefore wildly asymmetric, unlike every other curve in this article: flip the voltage sign and you do not get the mirror-image current. That asymmetry is the whole point — a diode is a valve, and the curve is the valve's specification sheet. (Why the junction behaves this way is semiconductor physics, deferred to the Electronics track; here the curve itself is the syllabus.)

The NTC thermistor. A semiconductor bead whose resistance falls steeply as temperature rises — heat frees extra charge carriers faster than lattice vibrations obstruct them, the opposite balance to a metal. Passing current self-heats the bead, so as the sweep proceeds the thermistor conducts ever more willingly and its $I$–$V$ curve bends toward the current axis. Thermistors are the standard temperature sensors of A-Level and IGCSE circuit questions (usually inside potential dividers, a story for the E3 chapter ahead).

Board level — convict the component from its data (CBSE/IGCSE/A-Level pattern)

A student measures two components. Component A: $(2.0\ \mathrm{V}, 0.25\ \mathrm{A})$, $(4.0\ \mathrm{V}, 0.50\ \mathrm{A})$, $(6.0\ \mathrm{V}, 0.75\ \mathrm{A})$. Component B: $(2.0\ \mathrm{V}, 0.10\ \mathrm{A})$, $(4.0\ \mathrm{V}, 0.32\ \mathrm{A})$, $(6.0\ \mathrm{V}, 0.75\ \mathrm{A})$. Classify each component, justify from the data, find A's resistance, and suggest what B could be.

Step 1. Ratio test on A (equation 3): $2.0/0.25 = 8.0\ \Omega$, $4.0/0.50 = 8.0\ \Omega$, $6.0/0.75 = 8.0\ \Omega$. Constant ratio ⇒ A is ohmic, $\boxed{R_A = 8.0\ \Omega}$.

Step 2. Ratio test on B: $2.0/0.10 = 20\ \Omega$, $4.0/0.32 = 12.5\ \Omega$, $6.0/0.75 = 8.0\ \Omega$. The ratio falls as the voltage rises ⇒ B is non-ohmic, with resistance decreasing at higher operating power.

Step 3. Name the suspect. Falling resistance under increasing self-heating is the signature of an NTC thermistor (a semiconductor whose warming frees more carriers). A filament lamp would show the opposite trend — ratios rising row by row.

Step 4. The marks-scheme sentences: for A, "V/I is constant, so the component obeys Ohm's law; the straight-line characteristic passes through the origin with gradient 1/8 A/V." For B, "V/I decreases with V, so resistance falls as current heats the component — thermistor-like behaviour." Justification from all rows, not one, is what the final mark buys.

Competitive level — static vs dynamic at the diode's knee (JEE-Main pattern, ~2.5 minutes)

A silicon diode's forward characteristic passes through $(0.68\ \mathrm{V}, 7.0\ \mathrm{mA})$, $(0.70\ \mathrm{V}, 10.0\ \mathrm{mA})$ and $(0.72\ \mathrm{V}, 14.0\ \mathrm{mA})$. At the operating point $0.70\ \mathrm{V}$: (a) find the static resistance; (b) estimate the dynamic resistance; (c) explain why the two differ so sharply.

Step 1. Static, from the coordinates (equation 2): $R = \dfrac{V}{I} = \dfrac{0.70}{10.0\times10^{-3}} = \boxed{70\ \Omega}$.

Step 2. Dynamic, from the changes across the neighbouring points: $r = \dfrac{\Delta V}{\Delta I} = \dfrac{0.72 - 0.68}{(14.0 - 7.0)\times10^{-3}} = \dfrac{0.04}{7.0\times10^{-3}} = \boxed{5.7\ \Omega}$.

Step 3. The gap explained: the operating point sits just past the knee, where the characteristic has become nearly vertical on an $I$–$V$ plot. The origin-to-point chord (static) still remembers the "dead" region below the knee — lots of volts spent for modest current — while the local tangent (dynamic) sees only the steep post-knee behaviour, where each hundredth of a volt buys milliamps. Chord and tangent are different lines on a curve; on the ohmic straight line they would be the same line and the two resistances would agree.

Step 4. Sanity check the direction: past the knee, static > dynamic (chord shallower than tangent). For a filament lamp at operating temperature the inequality flips — its curve bends the other way, so the tangent is shallower than the chord and dynamic > static. Knowing the direction of the inequality without computing is a free MCQ answer. Timing note: 2.5 minutes. Parts (a) and (b) are two divisions — the danger is unit slips (mA → A) and using one-sided differences; take the symmetric pair around the operating point as done here. Part (c) needs one sentence with the words "chord" and "tangent" — rehearse it once and it is permanent.

Axis conventions. The single biggest killer: on $I$–$V$ plots the ohmic gradient is $1/R$; on $V$–$I$ plots it is $R$. NCERT draws some figures $V$-vertical; CIE and most JEE papers draw $I$-vertical. Read the labels before judging steepness — an MCQ option list always includes the reciprocal answer.

"Steeper = more resistance" without checking axes. True on a $V$–$I$ plot, exactly false on an $I$–$V$ plot. Anchor the dictionary to one convention (I vertical: steep = willing = low R) and consciously flip when needed.

Extrapolating a static ratio. Computing $V/I$ at one point of a curve and applying it elsewhere (the diode warning above). Curved characteristics grant no extrapolation rights.

Diode symmetry error. Sketching the reverse branch as a mirror of the forward branch. The diode's whole identity is asymmetry: reverse current is essentially zero (microamps) until breakdown, which is beyond this syllabus.

Filament vs thermistor confusion. Both bend — in opposite directions. Metal heats → resists more → curve flattens ($I$-vertical). Semiconductor heats → conducts more → curve steepens. Tie each to its mechanism and the pair never swaps again.

Knee voltage forgotten. Treating a conducting silicon diode as a zero-volt wire. In circuit calculations past this module, a forward silicon diode costs about $0.7\ \mathrm{V}$; below that, it is open.

Board conventions. JEE-Main explicitly lists "V–I characteristics of Ohmic and non-ohmic conductors" — expect one every year (JEE-Advanced does not list it; the topic appears there only inside larger circuit problems, which is why this module carries no JA tag). AP Physics 2 phrases it as testing whether an element is ohmic from the slope of its I–V graph — its FRQs have asked students to design that experiment for a light bulb. IB (both SL and HL) names "ohmic and non-ohmic behaviour" with thermistors and LDRs in guidance. IGCSE and A-Level test curve sketching: practise drawing all four signatures freehand, with the knee near 0.7 V labelled.

Texts. NCERT Physics Class 12, Ch. 3 (V–I characteristics section) · HC Verma, Concepts of Physics Vol. 2 · CIE 9702 Topic 9.2 (I–V characteristics of resistor, lamp, diode) · IB Physics Theme B.5 (ohmic and non-ohmic behaviour).

Provenance. All results are standard. A verifier should recompute: Example 1's ratios ($8.0\ \Omega$ ×3 for A; $20$, $12.5$, $8.0\ \Omega$ for B); Example 2's static $0.70/0.010 = 70\ \Omega$ and dynamic $0.04/0.007 = 5.7\ \Omega$; the bulb numbers $R_{\text{hot}} = 230^2/100 = 529\ \Omega$, $I_{\text{op}} = 0.43\ \mathrm{A}$, cold $\approx 53\ \Omega$ giving inrush $\approx 4.3\ \mathrm{A}$.

Prerequisites. Ohm's Law & Resistance · Drift Velocity & the Microscopic Picture. Leads to. Temperature Dependence, Thermistors & Superconductors-Lite (coming in E3) · Semiconductors (future Electronics track).