Ohm's Law & Resistance

In 1827 a Cologne schoolteacher announced the simplest important equation in all of electricity — and was mocked for it. Georg Ohm claimed that the current through a metal wire is exactly proportional to the potential difference across it: double the push, double the flow, no exceptions, no drama. One education official sneered that the work was "a web of naked fancies"; Ohm lost his teaching post. It took the better part of a decade — and the British, who awarded him their highest medal — for physics to admit that the unemployed schoolteacher had handed it the master key to every circuit ever built. The law's power is its shape. Proportionality means one single number — the resistance $R$ — completely characterises a conductor: measure it once, at any convenient voltage, and you can predict the current at every voltage without touching the wire again. That is why $V = IR$ appears in every circuit calculation you will ever perform, from a two-mark board question to the power bus of a Mars rover. But the fine print is where exams live. Ohm's law is not a law of nature like conservation of energy — it is an experimental fact about some materials under some conditions, and the distinction between the always-true definition $R = V/I$ and the sometimes-true law " $R$ stays constant" is the single most-tested idea in this module. This article builds the law, its microscopic justification, the ohm itself, and the data-table craft that turns measurements into marks.

Key takeaways

Ohm's law: for a metallic conductor at constant temperature, $I \propto V$ — equivalently $V = IR$ with $R$ constant.
The ohm: $1\ \Omega = 1$ volt per ampere. Working multiples: $\mathrm{k\Omega} = 10^3\ \Omega$ , $\mathrm{M\Omega} = 10^6\ \Omega$ .
$R = V/I$ is a definition, computable for any device at any moment; Ohm's law is the extra experimental claim that this ratio does not change as $V$ changes.
The microscopic reason (from Drift Velocity & the Microscopic Picture): drift velocity is proportional to the field, so at fixed temperature current is proportional to voltage.
Conductance $G = 1/R$ (unit: siemens, S) measures how willingly a component conducts — large $G$ , easy flow.
Key idea: one measured number, $R$ , predicts the current at every voltage — for as long as the conductor itself (especially its temperature) stays unchanged.

The core insight

Ohm's law states that the ratio $V/I$ for a metallic conductor at fixed physical conditions is a constant — the resistance $R$ — so a single measurement of $R$ predicts the current at any potential difference; components whose $R$ shifts with temperature or operating point obey the definition $R = V/I$ at each moment but not the law.

Where this lives today

Every resistance measurement you have ever trusted is Ohm's law running in reverse. A digital multimeter on its Ω setting injects a small known current, reads the voltage that appears, and divides — Example 1, automated. Modern sensing goes one step further: most quantities that matter are converted into a resistance and then read out via $V = IR$ . A thermistor turns temperature into resistance for your thermostat; a strain gauge turns bridge flex into resistance for a structural monitor; your phone's battery-management chip watches the cell's internal resistance creep upward as the leading indicator of ageing. At the frontier, chip designers run "IR-drop analysis" across processors — literally this article's equation applied to billions of nanometre-scale copper paths at once, checking that the far corner of the die still receives its supply voltage. And the ohm itself is now defined through a quantum standard reproducible to parts per billion: a nineteenth-century schoolteacher's ratio, maintained by twenty-first-century quantum mechanics.

The physical picture

Water in a pipe is the picture every board answer is allowed to use. The pump pressure is the potential difference $V$ — the push per unit of stuff moved. The flow rate is the current $I$ . And the pipe's narrowness and internal roughness make up the resistance $R$ : the same pump drives a torrent through a wide smooth pipe and a trickle through a clogged one. Ohm's discovery, translated: for metal pipes of the electrical world, flow is exactly proportional to pressure. Double the battery voltage across a wire and precisely double the current flows. Why should the proportionality be exact? Drift Velocity & the Microscopic Picture already did the heavy lifting. Inside the metal, each free electron gains velocity from the field for a brief moment, loses it in a collision with a vibrating ion, and gains again — settling into a drift velocity proportional to the field: $v_d = (e\tau/m)E$ . Current is proportional to drift ( $I = nAv_d e$ ), and the field inside a uniform wire is proportional to the voltage across it. Chain the three proportionalities and $I \propto V$ drops out — provided the collision timetable $\tau$ stays fixed, which is precisely what "constant temperature" guarantees. Ohm's law is the macroscopic echo of a microscopic bargain: steady collisions, steady ratio.

Figure

The proportionality chain — voltage to field to drift to current

Hold on to what the chain does not promise. Nothing guarantees $\tau$ stays fixed if the wire heats itself, and nothing about a diode's junction resembles a metal's electron sea at all. Ohm's law has a jurisdiction — metallic conductors, steady temperature — and the entire next module, Ohmic & Non-Ohmic: I–V Characteristics, is a tour of the components that live outside it.

The law, the ohm, and the definition

The law first, in the form every mark scheme wants, with its condition attached — stating the condition is usually worth a mark on its own:

V = IR \qquad (1)

where: $V$ — potential difference across the conductor, in volts; $I$ — current through it, in amperes; $R$ — resistance, in ohms, constant for a given metallic conductor at constant temperature and physical conditions. Read $R$ as the conductor's stubbornness: how many volts of push it demands for each ampere it grants. The three rearrangements are one fact: $I = V/R$ predicts flow, $R = V/I$ extracts stubbornness, $V = IR$ prices the push. The unit deserves one honest sentence rather than a passing nod. One ohm is the resistance of a conductor through which a current of one ampere flows when one volt is applied:

1\ \Omega = \frac{1\ \mathrm{V}}{1\ \mathrm{A}} \qquad (2)

where: the symbol is the Greek capital omega. Real components span an enormous range — a metre of connecting wire is milliohms, lab resistors are tens to thousands of ohms, a voltmeter's innards are megohms — so the multiples $\mathrm{k\Omega}$ ( $10^3$ ) and $\mathrm{M\Omega}$ ( $10^6$ ) do daily duty, and unit-conversion slips around them cost real marks. Some boards (A-Level especially) also name the reciprocal. Conductance, $G = 1/R$ , measured in siemens (S), answers "how willingly?" instead of "how stubbornly?" — a $20\ \Omega$ resistor has $G = 0.05\ \mathrm{S}$ . Nothing new is being measured; it is the same information read from the other side, and it will streamline parallel-circuit algebra when Resistors in Series & Parallel arrives. Now the distinction that decides more marks than any other sentence in this chapter. The equation $R = V/I$ can be computed for any component at any operating point — a diode at 0.7 V, a glowing filament, anything. That is a definition, always legal, like defining average speed as distance over time. Ohm's law is a stronger, falsifiable claim: that for metallic conductors at steady conditions, this ratio comes out the same at every voltage. A filament lamp satisfies $V = IR$ at each instant — with a different $R$ each time. It obeys the definition and violates the law. When an examiner asks "does this component obey Ohm's law?", they are asking about the constancy, and the test is a graph, not a formula.

Derivation AWhat Ohm's law claims and why metals obey it

We establish exactly what is being claimed in equation (1) and derive its justification from the drift picture — the argument boards expect in "state and explain Ohm's law" answers. The strategy is to separate the always-true definition from the experimental claim, then chain the three proportionalities built in Drift Velocity & the Microscopic Picture. Watch the role of temperature: it enters as the guardian of the collision time τ, which is what makes "at constant temperature" part of the law's statement rather than decoration.

Step 1. Define resistance. For any conducting element, at any operating point, define $R \equiv V/I$ : the volts required per ampere delivered. This is a definition — always computable, never wrong, and by itself containing no physics.

Step 2. State the experimental fact. For a metallic conductor held at constant temperature, measurements (Ohm's own, and every school rig since) show the ratio $V/I$ is the same at every applied voltage. That constancy is Ohm's law. Symbolically: $I \propto V$ , with the constant ratio named $R$ .

Step 3. Justify it microscopically. In a uniform wire of length $L$ , the field is $E = V/L$ — proportional to $V$ . The drift velocity responds proportionally, $v_d = (e\tau/m)E$ , as long as the collision time $\tau$ is unchanged. The current follows the drift, $I = nAv_d e$ . Three proportional links: $V \to E \to v_d \to I$ , so $I \propto V$ . Why temperature guards the chain: τ is the average time between electron–ion collisions. Raise the temperature and the ions vibrate more violently, collisions come sooner, τ shrinks — the middle link of the chain changes strength, and the ratio V/I drifts. "Constant temperature" is precisely the condition "τ fixed."

Step 4. Extract the law's jurisdiction. The argument used the free-electron sea of a metal. Components without one — a diode's junction, an electrolyte, a gas discharge — never signed the bargain, and nothing entitles them to a constant ratio. They are governed by the definition of Step 1 alone. Verified: the chain reproduces the standard NCERT Ch. 3 argument; combining the three links gives $I = (nAe^2\tau/mL)\,V$ , i.e. constant $V/I$ at fixed τ, matching the Drude-model result quoted in UG treatments.

Measuring R: the rig and the craft

The circuit that certifies a component ohmic — and the one your practical exams use — is small and precise: a cell, a rheostat (variable resistor) to sweep the current, an ammeter in series with the component under test, and a voltmeter in parallel across it (the placement logic lives in Circuit Elements, Symbols & Diagrams). Slide the rheostat, record pairs of $(V, I)$ , and either plot the line or work the table.

Figure

The Ohm's law practical — sweep the rheostat, read V and I in pairs

The graph verdict: plotting $I$ (vertical) against $V$ (horizontal) — the convention most boards and this track use — an ohmic conductor gives a straight line through the origin, and the line's gradient is $1/R$ . Plot the axes the other way ( $V$ vertical) and the gradient is $R$ itself. Both conventions appear in real papers, which is why the first act of any graph question is to read the axis labels; the full craft of reading straight and curved characteristics — including every non-ohmic curve — is the business of I–V Characteristics, next. The table verdict matters just as much, because data arrive as numbers before they arrive as pictures. Compute $V/I$ for each row: if the ratio is constant within reading error, the component is ohmic and that constant is $R$ . If the ratio trends — rising steadily, say — the component is confessing that something (usually temperature) changed during the sweep. The competitive worked example below runs exactly this judgment.

Limiting cases

case	result	meaning
$R \to 0$	huge $I$ at any $V$	an ideal connecting wire — or a short circuit
$R \to \infty$	$I = 0$ at any $V$	an open switch, a broken filament
double $V$ , ohmic conductor	$I$ doubles	the defining test of the law
$V \to 0$	$I \to 0$ , line passes through origin	no push, no flow — a line missing the origin means a hidden emf, not a resistor
temperature rises during the sweep	$V/I$ ratio climbs between rows	the law's condition violated mid-experiment — metal wire heating up
$G = 1/R$	large $G$ = willing conductor	the same fact, read as conductance in siemens

Worked examples

Board level — resistance, prediction, and charge (CBSE/ICSE pattern, 3 marks)

A resistor connected across a $4.8\ \mathrm{V}$ battery carries a current of $0.20\ \mathrm{A}$ . (a) Find its resistance. (b) The battery is replaced by a $6.0\ \mathrm{V}$ supply. Assuming the resistor remains at the same temperature, find the new current. (c) How much charge passes through the resistor in 5.0 minutes at this new current?

Step 1. Part (a), from the definition: $R = \dfrac{V}{I} = \dfrac{4.8}{0.20} = \boxed{24\ \Omega}$ .

Step 2. Part (b): the same-temperature clause licenses Ohm's law — $R$ stays $24\ \Omega$ . So $I = \dfrac{V}{R} = \dfrac{6.0}{24} = \boxed{0.25\ \mathrm{A}}$ . Cross-check by proportionality: voltage rose by the factor $6.0/4.8 = 1.25$ , so current must rise from $0.20$ to $0.20 \times 1.25 = 0.25\ \mathrm{A}$ ✓.

Step 3. Part (c), reaching back to Electric Current & Charge Flow: $Q = It = 0.25 \times (5.0 \times 60) = \boxed{75\ \mathrm{C}}$ . (As electrons: $75/1.6\times10^{-19} \approx 4.7\times10^{20}$ .)

Step 4. The sentence that earns the method mark: "Since the conductor is metallic and its temperature is unchanged, $V/I$ is constant (Ohm's law), so the resistance found in (a) applies in (b)."

Competitive level — the data table with a fever (JEE-Main pattern, ~3 minutes)

A student sweeps a wire-wound resistor and records: $(0.5\ \mathrm{V}, 25\ \mathrm{mA})$ , $(1.0\ \mathrm{V}, 50\ \mathrm{mA})$ , $(2.0\ \mathrm{V}, 100\ \mathrm{mA})$ , $(3.0\ \mathrm{V}, 150\ \mathrm{mA})$ , $(4.0\ \mathrm{V}, 190\ \mathrm{mA})$ . (a) Find the resistance of the wire in its ohmic regime. (b) Explain the final reading. (c) What resistance does the wire present at the 4.0 V operating point?

Step 1. Convert and take ratios, row by row: $0.5/0.025 = 20$ , $1.0/0.050 = 20$ , $2.0/0.100 = 20$ , $3.0/0.150 = 20$ — four rows agree. $\boxed{R = 20\ \Omega}$ in the ohmic regime.

Step 2. Final row: $4.0/0.190 = 21.1\ \Omega$ . The ratio has risen — the current falls short of the $200\ \mathrm{mA}$ the straight line predicts. A wire's ratio rising at the high-power end is the signature of self-heating: at $4.0\ \mathrm{V}$ the wire dissipates the most power, its temperature climbs, collisions come faster, and the resistance grows. The law's constant-temperature condition failed, not the meters.

Step 3. Part (c): at the operating point, the definition still serves: $R = V/I = 4.0/0.190 = \boxed{21\ \Omega}$ (to two significant figures). One component, one moment, one honest ratio.

Step 4. The concept being tested: rows (a) used Ohm's law where its condition held; row (b) recognised the condition breaking; part (c) fell back to the definition, which never breaks. That three-way fluency is the whole module in one question. Timing note: about three minutes, nearly all of it division. Do the ratio test mentally in ascending order and stop computing the moment two rows disagree — that IS the answer to (b). Naming "temperature rise → resistance rise" without a story about collisions is enough for MCQ speed; save the mechanism for long-answer papers.

Exam traps

Dropping the condition. "State Ohm's law" answered without "provided temperature and other physical conditions remain constant" loses a mark on CBSE, ISC, IGCSE and A-Level papers alike. The condition is not a footnote; it is half the law.

Definition–law confusion. Computing $R = V/I$ for a diode is legal; concluding "so it obeys Ohm's law" is the classic error. Obedience = constancy of the ratio = straight line through the origin.

Gradient direction. On an $I$ – $V$ plot the gradient is $1/R$ ; on a $V$ – $I$ plot it is $R$ . Read the axes before the line. (Full graph craft, including curves: I–V Characteristics.)

Unit multiples. $\mathrm{mA}$ and $\mathrm{k\Omega}$ conspire: $V = IR$ with $I$ in mA and $R$ in kΩ conveniently gives $V$ in volts — but only if both prefixes are present. Mixed prefixes ( $\mathrm{mA}$ with $\Omega$ ) demand explicit conversion. When in doubt, work in A, V, Ω.

Line not through the origin. A straight $I$ – $V$ line with a nonzero intercept is not an ohmic resistor — an offset means something else in the circuit (an emf, a diode drop) is buried in the data. "Straight" and "through the origin" are two separate checks; examiners draw lines that pass one and fail the other.

Cold vs hot resistance. A question giving a component's resistance "at operating temperature" must not be answered with room-temperature reasoning, and vice versa. If the wire heats mid-problem, Ohm's law connects readings only within each temperature, never across.

Constants and conventions. All boards here accept $V = IR$ with SI units; IGCSE expects recall-and-use with the ohm defined as volt per ampere; A-Level additionally uses conductance $G = 1/R$ in siemens. No board in this module needs $g$ , but as always: use exactly the constants the paper supplies.

Origins · 1 minGeorg Ohm and how this was discovered

Georg Ohm

1789–1854 · Cologne and Munich

Georg Ohm did precision physics on a schoolteacher's salary with home-built apparatus. His masterstroke was recognising that the batteries of his day were too unsteady for careful measurement, so he switched to a thermocouple — two metal junctions held at different fixed temperatures — which gave a rock-steady source, and he read currents from the twist of a magnetised needle. Sweeping through wires of different lengths and thicknesses, he extracted the proportionality $V = IR$ and published it in 1827 in Die galvanische Kette. The German establishment savaged the book — too mathematical, too abstract — and Ohm resigned his post in the fallout. Britain's Royal Society rehabilitated him with the Copley Medal in 1841, and in 1881 the international electrical congress fixed his name to the unit of resistance itself: a better verdict than "a web of naked fancies."

History

1781

— Henry Cavendish anticipates the proportionality using his own body as the meter, grading currents by the strength of the shock — and never publishes.

1827

— Ohm's Die galvanische Kette states V = IR; the hostile reception costs him his teaching position.

1841

— The Royal Society awards Ohm the Copley Medal; the law finally crosses into mainstream engineering.

1881

— The Paris Electrical Congress names the unit of resistance the ohm.

1900

— Drude's collision model of the electron gas supplies the microscopic why behind the proportionality.

Going further

Texts. NCERT Physics Class 12, Ch. 3 (Current Electricity) · HC Verma, Concepts of Physics Vol. 2, Ch. 32 · CIE 9702 Topic 9.2 · IGCSE 0625 Section 4.2 (recall and use V = IR).

Provenance. All results are standard. A verifier should recompute: Example 1's $R = 4.8/0.20 = 24\ \Omega$ , $I = 6.0/24 = 0.25\ \mathrm{A}$ , $Q = 0.25 \times 300 = 75\ \mathrm{C}$ , electron count $4.7\times10^{20}$ ; Example 2's ratios $20\ \Omega$ (×4 rows) and $4.0/0.190 = 21.1\ \Omega$ ; conductance $G = 1/20 = 0.05\ \mathrm{S}$ .

Prerequisites. Electric Current & Charge Flow · Circuit Elements, Symbols & Diagrams · Drift Velocity & the Microscopic Picture. Leads to. Ohmic & Non-Ohmic: I–V Characteristics · Resistivity & Geometry (coming in E3) · Ohm's Law and Conductivity (UG deep dive).

Is Ohm's law a real law of physics if so many things disobey it?

It is a law in the older, experimental sense: a reliable regularity with a stated jurisdiction — metallic conductors, steady temperature — rather than a universal principle like charge conservation. Within its jurisdiction it is astonishingly exact (precision resistors hold their ratio to parts per million). Outside it, no one ever promised anything: diodes, electrolytes and filaments simply aren't parties to the bargain. Physicists keep calling it a law partly from history and partly from respect: one measured number predicting a component's behaviour at every voltage is as law-like as electricity gets.

Why does temperature get singled out in the statement of the law?

Because temperature controls the one microscopic knob the proportionality depends on: the average time τ between electron–ion collisions. The chain $V \to E \to v_d \to I$ has fixed strength only while τ is fixed, and τ shortens when the lattice vibrates harder — that is, when the wire warms. Other "physical conditions" (mechanical strain, illumination for some materials) can also shift the ratio, which is why careful statements say "temperature and other physical conditions." Temperature is named because it is the condition that actually varies in practice — often because the current itself heats the wire.

If I double the voltage, the wire carries more current, which heats it more. Doesn't Ohm's law then break itself?

In principle, yes — and good exam questions exploit exactly this. A wire dissipating noticeable power warms up, its resistance creeps upward, and the I–V line begins to sag below proportionality. In practice, school-lab currents through a proper resistor produce negligible heating, so the law holds beautifully; the effect only bites in components designed to run hot (filaments) or in sweeps pushed to high power (the final row of Worked Example 2). This self-heating boundary is precisely where this module hands over to I–V Characteristics.

Which is more fundamental — V = IR or the microscopic drift picture?

The drift picture. $V = IR$ is what the collision physics looks like from outside the wire: proportional response, summarised in one constant. The microscopic account (drift proportional to field, collisions resetting the gain) explains both the law and its exceptions — heat the lattice and the ratio changes; replace the electron sea with a semiconductor junction and proportionality never appears. Exams reflect the hierarchy: MCQs test the formula, but the long-answer "explain why metals obey Ohm's law" wants the chain of proportionalities from the drift picture.

What is conductance actually for? It feels like resistance's redundant twin.

For a single component it is redundant — $G = 1/R$ contains identical information. Its value appears in networks: when components sit in parallel, conductances simply add, while resistances combine by the awkward reciprocal rule. Electricians and chip designers, who parallel things constantly, often think in siemens for that reason. At this stage you need to recognise the definition, the unit, and the reading "big G = willing conductor"; the algebraic payoff arrives with Resistors in Series & Parallel.

My multimeter shows a lamp's resistance as 30 ohms, but the lamp's rating implies ten times that. Which is wrong?

Neither — you measured a cold filament. The meter's tiny test current leaves the filament at room temperature, where its resistance is genuinely small. At operating brightness the filament runs thousands of degrees hotter and its resistance is several times larger. The two readings disagree because resistance is temperature-dependent, and the lamp spends its life far from room temperature. This is the everyday proof that a filament is non-ohmic — the ratio V/I depends on the operating point. The curve connecting those two states is drawn and dissected in I–V Characteristics.

Do I write Ohm's law as V = IR or I = V/R in exams?

Any algebraically correct form earns the mark — but the statement of the law should be in words with the condition: current through a metallic conductor is proportional to the potential difference across it, provided temperature and other physical conditions remain constant. Then deploy whichever rearrangement the calculation wants. One presentation habit worth keeping: quote the formula, substitute with units visible, then state the answer with its unit — mark schemes for CBSE, ISC and IGCSE award working, not just the final number.