Resistance and Ohm's Law

Spec mapping: OCR H556 Module 4.2 — Energy, power and resistance (definition of resistance $R = V/I$R=V/I in ohms; Ohm's law as the empirical statement that current is proportional to pd at constant temperature; distinction between ohmic and non-ohmic components; foundations for I-V characteristics in lesson 6 and resistivity in lesson 7). Refer to the official OCR H556 specification document for exact wording.

We now have current ($I$I), charge ($Q$Q), drift velocity ($v$v), EMF ($\varepsilon$ε) and potential difference ($V$V) all clearly defined. It is time to link them together with the single most famous equation in circuit analysis: $V = IR$V=IR, often called Ohm's law.

But "Ohm's law" is not just "$V = IR$V=IR". The equation $V = IR$V=IR is simply the definition of resistance ($R = V/I$R=V/I, rearranged), and it applies to any component — ohmic or not. Ohm's law itself is a statement about a particular class of materials — ohmic conductors — whose resistance stays constant regardless of the current. Confusing the definition with the law is the single most common A-Level electricity mistake, and OCR mark schemes are unforgiving.

This lesson fixes that confusion once and for all. We define resistance precisely, state Ohm's law in its careful empirical form, distinguish ohmic from non-ohmic behaviour, and develop the microscopic reasoning (drift velocity, lattice scattering) that explains why metallic resistance rises with temperature. By the end you should be able to identify ohmic versus non-ohmic data at a glance and to articulate the difference between an equation that defines and an equation that asserts a physical law.

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Defining Resistance

The electrical resistance $R$R of a component is defined as the ratio of the potential difference $V$V across it to the current $I$I flowing through it:

$R = \frac{V}{I}$R=IV​

• $R$R is measured in ohms ($\Omega$Ω), named after Georg Ohm (1789–1854).
• $1 \, \Omega = 1 \text{ V} / 1 \text{ A} = 1$1Ω=1 V/1 A=1 V A$^{-1}$−1.

Rearranged, this becomes the familiar $V = IR$V=IR form used in almost every calculation. A third form, $I = V/R$I=V/R, is useful when you want to predict the current drawn by a known resistor at a given pd.

Critical: This is a definition. It is just the labelling of the ratio $V/I$V/I with the letter $R$R. It does not assert that $R$R is constant — only that, at any operating point ($V, I$V,I), you can compute the resistance at that point as $V/I$V/I. For an ohmic wire $R$R is the same at every operating point; for a non-ohmic component (filament lamp, diode, thermistor) $R$R varies as $V$V and $I$I vary, but you can still calculate it locally.

Worked Example 1 — Predict current from R

A resistor of $220$220 $\Omega$Ω has a pd of $6.0$6.0 V across it. Calculate the current.

$I = \frac{V}{R} = \frac{6.0}{220} \approx 0.0273 \text{ A} \approx 27 \text{ mA}$I=RV​=2206.0​≈0.0273 A≈27 mA

Worked Example 2 — Determine R from V and I

A current of $0.50$0.50 A flows through a wire when a pd of $12$12 V is applied across it. Calculate its resistance.

$R = \frac{V}{I} = \frac{12}{0.50} = 24 \text{ }\Omega$R=IV​=0.5012​=24 Ω

The wire is not necessarily ohmic — we have only one $(V, I)$(V,I) pair. To establish whether $R = 24$R=24 $\Omega$Ω at all operating points, we'd need to vary $V$V and check that $V/I$V/I remains constant.

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Ohm's Law — The Actual Statement

Ohm's Law: The current through a metallic conductor is directly proportional to the potential difference across it, provided the temperature (and other physical conditions) remains constant.

In symbols:

$I \propto V \qquad \text{(at constant temperature)}$I∝V(at constant temperature)

which is equivalent to saying $R = V/I$R=V/I is a constant, independent of $V$V or $I$I. Components that satisfy Ohm's law are called ohmic conductors. A metal wire held at constant temperature is the classic example.

If you draw a graph of $I$I (vertical) against $V$V (horizontal) for an ohmic conductor, you get a straight line through the origin. The gradient is $1/R$1/R, and equivalently $R = V/I$R=V/I is the inverse gradient.

Exam Tip: OCR mark schemes are strict on the phrase "at constant temperature" (or equivalently "provided physical conditions are constant"). Leaving it out can cost a mark even if the rest of the statement is correct. Ohm's law is not universal; it's a property of certain materials under certain conditions.

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The Crucial Distinction — Definition vs Law

Statement	Status	Applies to
$R = V/I$R=V/I	Definition of resistance	Any component, at any operating point
$I \propto V$I∝V at constant $T$T (Ohm's law)	Physical law about a class of materials	Only ohmic conductors

Every component has a resistance at any chosen operating point — you can always compute $R = V/I$R=V/I. But only ohmic components have a constant $R$R across their full operating range. For non-ohmic components (filament lamp, diode, thermistor), $R$R varies with $V$V (or with $T$T, which varies with $V$V via $I^2 R$I2R heating).

Put another way: Ohm's law is the experimental observation that for some materials, a graph of $I$I against $V$V is a straight line through the origin. For others — diodes, lamps, thermistors — the graph is curved, and those materials are non-ohmic.

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Non-Ohmic Behaviour — A Preview

The next lesson examines the I-V characteristics of five canonical components:

• Ohmic resistor — straight line through the origin.
• Filament lamp — S-curve, because the filament heats up and its resistance rises.
• Diode (incl. LED) — strongly non-linear, switches on at about $0.6$0.6–$0.7$0.7 V (silicon) or $2$2–$3$3 V (LEDs).
• NTC thermistor — curve, because carrier density $n$n rises with $T$T (lesson 2).
• LDR — resistance drops in bright light (photons free carriers).

For any of these, $V = IR$V=IR still gives the correct resistance at a particular operating point — but $R$R is not a constant. So if you're asked to compute $R$R for a filament lamp at a specific $V$V and $I$I, you can still use $R = V/I$R=V/I — you just cannot assume the answer applies at a different $V$V.

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Decision Tree — Ohmic or Non-Ohmic?

flowchart TD
  A["Collect I-V data over a range of V"] --> B["Plot I (y-axis) against V (x-axis)"]
  B --> C{"Straight line through origin?"}
  C -->|Yes| D["OHMIC. R = V/I = constant = 1/gradient"]
  C -->|"No, curved or kinked"| E["NON-OHMIC. R varies with operating point."]
  E --> F["Compute R = V/I at the specific operating point"]
  F --> G["Do NOT extrapolate R to other voltages without re-measuring"]

This decision tree is the workflow for any I-V data set at A-Level.

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Worked Example 3 — Straight-Line I-V (Ohmic)

The following readings are taken for a metal wire at constant temperature.

$V$V (V)	$I$I (A)	$V/I$V/I ($\Omega$Ω)
0	0	—
2.0	0.10	20
4.0	0.20	20
6.0	0.30	20
8.0	0.40	20

Is the wire ohmic? What is its resistance?

• $I$I is directly proportional to $V$V (constant ratio $V/I = 20$V/I=20 $\Omega$Ω).
• Plot is a straight line through the origin (gradient $1/R = 0.05$1/R=0.05 A V$^{-1}$−1).
• The wire is ohmic, $R = 20$R=20 $\Omega$Ω at every operating point shown.

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Worked Example 4 — Curved I-V (Non-ohmic Filament Lamp)

A filament lamp gives the following readings.

$V$V (V)	$I$I (A)	$R = V/I$R=V/I ($\Omega$Ω)
0.5	0.10	5.0
1.0	0.16	6.3
2.0	0.24	8.3
4.0	0.36	11.1
6.0	0.45	13.3

The ratio $V/I$V/I increases with $V$V. The lamp is non-ohmic: its resistance rises as more current flows, because the filament is heating up. At $0.5$0.5 V, $R = 5.0$R=5.0 $\Omega$Ω; at $6.0$6.0 V, $R = 13.3$R=13.3 $\Omega$Ω — almost triple. The plot of $I$I against $V$V would curve concave-down — pulling progressively further below a straight line as $V$V increases.

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Temperature Dependence of Metals — Microscopic Explanation

Why does the filament's resistance rise with temperature? Microscopically (from lesson 2's $I = nAvq$I=nAvq picture):

• A metal wire conducts because the "sea" of delocalised free electrons drifts under the applied field.
• As temperature rises, the positive ions of the lattice vibrate more violently.
• Electrons are scattered more often by the vibrating ions; their mean free path shortens.
• Each scattering event reduces the mean drift velocity $v$v, so at fixed $V$V, the current $I = nAvq$I=nAvq falls.
• Equivalently, $R = V/I$R=V/I rises.

For pure metals over a wide range:

$R(T) \approx R_0 (1 + \alpha \Delta T)$R(T)≈R0​(1+αΔT)

where $\alpha$α is the temperature coefficient of resistance ($\sim +0.004$∼+0.004 K$^{-1}$−1 for copper). You're not expected to memorise this formula at A-Level, but you must be able to explain qualitatively why metallic $R$R rises with $T$T.

Contrast this with a semiconductor (thermistor), where rising $T$T exponentially increases the number density $n$n of free charge carriers (electrons gaining enough thermal energy to jump the band gap), and $R$R falls. Both behaviours follow from $I = nAvq$I=nAvq:

• Metal: $n$n constant, $v$v falls with $T$T $\Rightarrow$⇒ $R$R rises with $T$T.
• Semiconductor: $n$n rises rapidly with $T$T, swamping the smaller $v$v decrease $\Rightarrow$⇒ $R$R falls with $T$T.

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The Graph Convention — Gradient = 1/R, Not R

Most textbooks (and OCR exam questions) plot I-V curves with $V$V on the horizontal axis and $I$I on the vertical axis. With this convention:

• Gradient $= \Delta I / \Delta V = 1/R$=ΔI/ΔV=1/R.
• For an ohmic component, a steeper line means smaller $R$R.
• For a non-ohmic component, the local gradient is $dI/dV$dI/dV at that operating point — not the same as $1/R = I/V$1/R=I/V from the chord through the origin.

The chord from the origin to a point $(V_0, I_0)$(V0​,I0​) has slope $I_0/V_0 = 1/R(V_0)$I0​/V0​=1/R(V0​) — the secant gradient = $1/R$1/R at $V_0$V0​.

The tangent at $(V_0, I_0)$(V0​,I0​) has slope $dI/dV$dI/dV — the differential conductance, not the same as $1/R$1/R for a non-ohmic component.

For ohmic resistors these two coincide. For non-ohmic components they generally differ — and OCR exam questions will catch you out by asking for "the resistance at $V = 4$V=4 V" (chord) versus "the gradient of the I-V curve at $V = 4$V=4 V" (tangent).

Some textbooks swap the axes ($V$V on vertical, $I$I on horizontal). In that case the gradient is $R$R itself (not $1/R$1/R). Always check which convention a question is using.

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Worked Example 5 — Thermistor

A thermistor (NTC type) has a resistance of $4500$4500 $\Omega$Ω at $20$20 $^{\circ}$∘C and $750$750 $\Omega$Ω at $80$80 $^{\circ}$∘C. A pd of $6.0$6.0 V is applied across it (with negligible self-heating). Calculate the current at each temperature.

At $20$20 $^{\circ}$∘C: $I = V/R = 6.0 / 4500 = 1.33 \times 10^{-3}$I=V/R=6.0/4500=1.33×10−3 A $\approx 1.3$≈1.3 mA.

At $80$80 $^{\circ}$∘C: $I = V/R = 6.0 / 750 = 8.0 \times 10^{-3}$I=V/R=6.0/750=8.0×10−3 A $= 8.0$=8.0 mA.

Raising the temperature by $60$60 K raises the current by a factor of $6$6. This dramatic response is what makes thermistors useful as temperature sensors in potential-divider circuits (lesson 12).

Note: at each operating point, $V = IR$V=IR holds (it's a definition). But $R$R is different at $20$20 $^{\circ}$∘C and $80$80 $^{\circ}$∘C, so the thermistor is not ohmic.

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Worked Example 6 — Distinguishing Definition from Law in a Mark-Scheme Answer

A student writes: "Ohm's law states that V = IR."

What's wrong with this answer? Re-write it correctly.

Wrong because: $V = IR$V=IR is the definition of resistance (or its rearrangement), not Ohm's law. It applies to any component, including non-ohmic ones (where $R$R varies with the operating point). It is not the empirical law that Ohm discovered.

Correct statement: "Ohm's law states that the current through an ohmic conductor is directly proportional to the potential difference across it, provided the temperature and physical conditions remain constant: $I \propto V$I∝V, equivalently $R = V/I$R=V/I is constant."

OCR has dropped marks routinely for this confusion. Be precise.

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Worked Example 7 — Microscopic Origin of Resistance from $I = nAvq$I=nAvq