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Resistance, Resistivity & Potential Dividers

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Edexcel IAL Physics · Unit — Electricity

Resistance, Resistivity & Potential Dividers

Electrons moving through a material keep bumping into stuff — how much they bump into, and how the wire's shape and material affect that, determines resistance. Wire that up in a smart way (a potential divider) and you can split a voltage into exactly the piece you want, or build a sensor that reacts to light or heat.

Quick Summary

Resistance is caused by free electrons colliding with fixed ions as they drift through a conductor, transferring kinetic energy as heat.
Resistivity (ρ) is a property of the material itself (not its shape) — it tells you how strongly a material opposes current flow. R = ρL/A links resistance to length, area, and resistivity.
Core Practical 7 finds resistivity by measuring resistance at different wire lengths, plotting R vs L, and using the gradient (= ρ/A).
Drift velocity is the average speed of charge carriers, linked to current by I = nqvA.
Potential difference is energy transferred (or work done) per unit charge, measured with a voltmeter connected in parallel.
Potential dividers split a supply voltage between two series resistors in the ratio of their resistances: V_out = [R₂ / (R₁+R₂)] × V_in.
Potentiometers, LDRs and thermistors turn potential dividers into variable-voltage or sensor circuits (light and temperature sensing).

1. Electrical Resistivity

Why does resistance exist at all?

Picture a wire as a crowded corridor full of people standing still (these are the fixed metal ions) while a stream of people (the free electrons) try to rush through to the other end. They can't move in a straight line — they keep bumping into the standing people, bouncing off in random directions, and losing a bit of energy (as heat) every time they collide. That's what's happening inside a metal wire carrying current.

Since current is just the flow of charge, anything that gets in the way of that flow — i.e. the ions resisting the electrons — is what we call resistance. The more collisions there are, the more resistance there is, and the more electrical energy gets converted into heat (this is why wires and resistors get warm).

➜ ELECTRONS ➜ − (•)(•)(•)(•)(•)(•) + − (•)(•)(•)(•)(•)(•) + (•) = metal ion (fixed, in the way) − (•)(•)(•)(•)(•)(•) + • = electron (small, darting between them) Electrons zig-zag past ions, losing energy on every collision → this IS resistance.

What resistance actually depends on

Resistance isn't just about the material — it's also about the wire's shape. Three things matter:

Length (L): a longer wire means more ions to crash into on the way through → more resistance.
Cross-sectional area (A): a thicker wire gives electrons more "lanes" to travel down at once → less resistance.
Resistivity (ρ): a property of the material itself, independent of its shape, that says how strongly it opposes current.

R = ρL / A

Resistance = (resistivity × length) ÷ cross-sectional area

R = resistance (Ω)

ρ = resistivity of the material (Ω m)

L = length of the conductor (m)

A = cross-sectional area (m²)

Think of it like a motorway

Doubling the length of the wire = doubling the resistance (twice as many "obstacles" to get through). Doubling the cross-sectional area = halving the resistance (twice as many lanes for the traffic). A thick, short wire = low resistance. A thin, long wire = high resistance.

Resistivity of common materials (room temperature)

Category	Material	Resistivity ρ / Ω m
Metals	Copper	1.7 × 10⁻⁸
	Gold	2.4 × 10⁻⁸
	Aluminium	2.6 × 10⁻⁸
Semiconductors	Germanium	0.6
Semiconductors	Silicon	2.3 × 10³
Insulators	Glass	10¹²
Insulators	Sulfur	10¹⁵

Notice the range — from 10⁻⁸ up to 10¹⁵. That's why copper wraps around every wire in your house (barely resists current) while rubber and plastic coat the outside (essentially block it completely). Resistivity also depends on temperature — it's not a fixed constant for a material at all temperatures.

Worked Example

Two cylinders — one copper (diameter 5 mm, length 8 mm), one aluminium (diameter 10 mm, length 16 mm). Copper ρ = 1.7 × 10⁻⁸ Ω m, Aluminium ρ = 2.6 × 10⁻⁸ Ω m. Which is the better conductor?

Step 1: Better conductor = lower resistance, so calculate R = ρL/A for both, using A = π(d/2)².

Copper: A = π×(2.5×10⁻³)² = 2.0×10⁻⁵ m² → R = (1.7×10⁻⁸ × 8×10⁻³) / 2.0×10⁻⁵ = 6.8 × 10⁻⁶ Ω

Aluminium: A = π×(5×10⁻³)² = 7.9×10⁻⁵ m² → R = (2.6×10⁻⁸ × 16×10⁻³) / 7.9×10⁻⁵ = 5.3 × 10⁻⁶ Ω

Since the aluminium cylinder has the lower resistance, the aluminium cylinder is the better conductor — even though copper has lower resistivity! Its bigger cross-sectional area more than makes up for it. This is a classic exam trap: don't just compare ρ values, you must calculate R.

Practice Question 1

A copper wire of length 1.5 m and diameter 0.40 mm has resistivity 1.7 × 10⁻⁸ Ω m. Calculate its resistance.

Practice Question 2

If the diameter of a wire is doubled, but its length and material stay the same, what happens to its resistance? Explain why.

2. Core Practical 7 — Investigating Resistivity

Aim: determine the resistivity of a length of wire (e.g. constantan).

Variables

Independent variable: length, L, of the wire (m)
Dependent variable: the current, I, through the wire (A)
Control variables: voltage across the wire; the material the wire is made from

Equipment & resolution

Equipment	Purpose	Resolution
Ammeter	Measures current through the wire	0.01 A
Voltmeter	Measures voltage across the wire	0.1 V
2.0 m constantan wire (22–36 swg)	The test wire	—
Flying lead	Crocodile clip to connect at any point along the wire	—
Metre ruler	Measures wire length	1 mm
Micrometer	Measures wire diameter	0.01 mm
Power supply	Provides the voltage	—

Method (the exam-safe version)

Measure the wire's diameter with a micrometer at 5–10 random points along it, and calculate the mean.
Tape or clamp the wire to a ruler so one end (attached to the circuit) is at the 0 cm mark — this avoids a zero error.
Connect an ammeter in series and a voltmeter in parallel with the wire. Attach the flying lead at 0.25 m, set the power supply to 6.0 V.
Read the current, then switch off immediately to prevent the wire heating up (which would change its resistivity).
Repeat at 0.25 m intervals up to the full 2.0 m length (8 readings), taking 3 repeats at each length and averaging.
Calculate resistance at each length using R = V/I.
Plot a graph of length (L) vs resistance (R), draw a line of best fit, and find the gradient.

gradient = ΔR/ΔL = ρ/A → ρ = gradient × A

Comparing R = ρL/A with y = mx: y = R, x = L, so the gradient of the R–L graph equals ρ/A

Why use a graph instead of one reading?

Using the gradient of a graph (rather than one single R and L value) averages out random errors across many data points, giving a far more reliable value for ρ than any single measurement could.

Evaluating the experiment

Systematic errors: the end of the wire attached to the circuit must genuinely start at the 0 mark on the ruler, otherwise every length measurement is offset (a zero error).

Random errors and how to reduce them:

Only allow small currents to flow — resistivity depends on temperature, and current flowing through the wire heats it up, which would change ρ mid-experiment.
Switch the current off between readings so the wire doesn't build up heat.
Take 5–10 diameter measurements with the micrometer and average them, since the wire's thickness may not be perfectly uniform.

Safety: a thin wire carrying high current gets very hot — never touch it while the circuit is switched on; switch off immediately if you smell burning; keep liquids away from the equipment (damage + short circuit risk).

Worked Example

A student measures average diameter = 0.19 mm, and gets a graph of R against L with gradient 15.71 Ω/m (found from two points: (0.3, 5.00) and (1.7, 27.00)).

Step 1: Cross-sectional area: A = π(0.191×10⁻³)²/4 = 2.87 × 10⁻⁸ m²

Step 2: ρ = gradient × A = 15.71 × 2.87×10⁻⁸ = 4.51 × 10⁻⁷ Ω m

Practice Question

Why must the current be switched off between readings in this experiment, and why does it matter for the value of resistivity obtained?

3. Calculating Current & Drift Velocity

What is drift velocity?

Free electrons in a wire don't shoot straight through like a bullet — they zig-zag chaotically, bouncing off ions in random directions. But underneath all that randomness, there's a slow, steady overall drift in one direction (caused by the electric field pushing them). That average net speed is the drift velocity, and it's surprisingly slow — typically around 10⁻³ m s⁻¹ (a fraction of a millimetre per second!). Compare that to how "instantly" a lamp switches on when you flip a switch — that's because there are charge carriers throughout the wire that they all start moving together almost immediately, even though each individual electron drifts slowly.

Analogy: a full hosepipe

Imagine a hosepipe already completely full of water. The moment you turn the tap, water comes out the far end instantly — not because that water travelled the length of the hose in an instant, but because the whole column of water moved together at once. Electrons in a wire behave the same way.

Positive vs negative charge carriers

Current is always defined as flowing in the same conventional direction (from + to −), regardless of what's actually carrying the charge:

If charge carriers are positive, drift velocity points in the same direction as the current.
If charge carriers are negative (as in a metal — electrons), drift velocity points in the opposite direction to the current.

I = nqvA

Current = (number density) × (charge) × (drift velocity) × (cross-sectional area)

I = current (A)

n = number density of charge carriers (m⁻³)

q = charge on each carrier (C)

v = drift velocity (m s⁻¹)

A = cross-sectional area of the wire (m²), found using A = πr²

What the equation is telling you:

v is inversely proportional to n — the more charge carriers packed into a given volume, the slower each one needs to drift to carry a given current (think of a wide, slow-moving crowd vs a narrow, fast-moving one carrying the same "flow rate" of people).
I is directly proportional to n — more available charge carriers per unit volume means more current for the same drift speed.
This explains why insulators (very low n) barely conduct at all — even with a huge electric field pushing them, there just aren't enough charge carriers to produce meaningful current.

Worked Example

Copper wire: number density of conduction electrons n = 9.2 × 10²⁸ m⁻³, current I = 3.5 A, cross-sectional area A = 1.5 mm². Find the average drift velocity.

Step 1: Rearrange I = nqvA for v: v = I / (nqA)

Step 2: Convert A = 1.5 mm² = 1.5 × 10⁻⁶ m². Use q = 1.60 × 10⁻¹⁹ C (charge of an electron).

Step 3: v = 3.5 / [(9.2×10²⁸) × (1.60×10⁻¹⁹) × (1.5×10⁻⁶)]

v ≈ 0.16 mm s⁻¹ (2 s.f.) — genuinely slower than a snail!

Unit trap

A is in m², v is in m s⁻¹, and n is in m⁻³. Diameters and areas are almost always given in mm or mm² in exam questions — always convert to metres/m² substituting into I = nqvA, or your answer will be out by a factor of a million (or more)!

Practice Question

A silver wire of cross-sectional area 2.0 mm² carries a current of 2.4 A. The number density of free electrons in silver is 5.8 × 10²⁸ m⁻³. Calculate the drift velocity of the electrons. (charge of electron = 1.60 × 10⁻¹⁹ C)

Why resistivity varies so wildly between materials

Since I ∝ n (number of charge carriers), resistivity is really a story about how many mobile charge carriers a material has:

Conductors (metals): huge number of free electrons per unit volume → low resistivity.
Insulators: essentially no free charge carriers (n ≈ 0) → extremely high resistivity, virtually no current flows regardless of voltage.
Semiconductors (e.g. silicon): fewer free electrons than a metal, but the number of free electrons increases with temperature — this is why a semiconductor's resistivity actually drops as it gets hotter (opposite to a metal).

4. Potential Difference & Conductor Length

What is potential difference, really?

A cell makes one end of a circuit positive and the other negative — this sets up a potential difference (p.d.) across the circuit. Formally:

Definition

Potential difference across a component = the energy transferred (or work done) per unit charge passing through it.

V = W / Q

Potential difference = Work done ÷ Charge

V = potential difference (V, equivalent to J C⁻¹)

W = work done / energy transferred (J)

Q = charge (C)

In a series circuit, the p.d. of the power supply is always shared between all the components — the individual p.d.s add up to equal the supply e.m.f. A voltmeter measures p.d. and must always be connected in parallel with the component you're measuring (never in series — it would disrupt the circuit).

Connecting resistivity to conductor length

We already know R = ρl/A, so for a uniform conductor at constant temperature, resistance increases as length increases. Combine that with Ohm's Law:

V = IR

Potential difference = Current × Resistance

Since resistance increases uniformly with length, and V = IR (with I constant in a series circuit), potential difference across the wire also increases uniformly with length. This is exactly why the resistivity practical works: a longer section of wire has both higher resistance a bigger share of the potential difference.

Practice Question

A uniform wire carries a constant current of 0.40 A. If the p.d. across a 0.50 m length of the wire is 1.8 V, what would the p.d. be across a 1.5 m length of the same wire (assuming constant temperature)?

5. Potential Dividers

The core idea

The electrical voltages rule (Kirchhoff's 2nd law, though you don't need that name) says:

Rule

The sum of the e.m.f.s in a closed circuit loop is equal to the sum of the potential differences around that loop.

So when two resistors R₁ and R₂ are connected in series across a supply V_in, the supply voltage gets divided between them — a potential divider. This is deliberately used to produce a chosen fraction of the input voltage as an output, V_out, taken across just one of the resistors.

Potential dividers have three main uses:

To provide a variable potential difference
To enable a specific chosen potential difference
To split a supply's p.d. between two or more components

Vin ─┬── R1 ──┐ │ │ (+) Vout (measured across R2) │ │ └── R2 ──┴─── 0V

V_out = [ R₂ / (R₁ + R₂) ] × V_in

Output voltage = (fraction of total resistance that R₂ makes up) × input voltage

V_out = output voltage, measured across R₂ (V)

V_in = input/supply voltage (V)

R₁, R₂ = the two series resistors (Ω)

The resistor with the bigger resistance always gets the bigger share of the p.d. — this follows directly from V = IR, since the current I is the same through both resistors (they're in series).

V₁ / V₂ = R₁ / R₂

The ratio of the p.d.s across each resistor equals the ratio of their resistances (also: V₁ = IR₁ and V₂ = IR₂)

Intuition check

Increase R₁'s resistance → it "hogs" more of the voltage, so V across R₁ increases and V across R₂ (=V_out) decreases. The two p.d.s must always add back up to V_in — if one goes up, the other must come down by exactly the same amount.

Worked Example

A potential divider circuit has R₁ = 20 kΩ and R₂ = 12 kΩ, designed to light a lamp when V_out reaches 5.3 V. Find V_in.

Step 1: Start from V_out = [R₂/(R₁+R₂)] × V_in

Step 2: Rearrange: V_in = V_out × [(R₁+R₂)/R₂]

Step 3: V_in = 5.3 × (32/12) = 5.3 × 2.667 = 14.13...

Wait — check with the actual numbers given: V_in = 5.3 × [(20+12)/20] = 5.3 × 1.6 = 8.48 V ≈ 8.5 V (using R₁ in the denominator here because in this specific circuit, V_out was taken across R₁ — always check which resistor V_out is measured across before substituting!)

Classic mistake

Students often plug numbers into the formula without checking which resistor V_out is actually measured across. The formula V_out = [R₂/(R₁+R₂)]×V_in assumes V_out is across R₂ — if your diagram has V_out across the resistor instead, you must swap which resistance goes on top of the fraction!

Practice Question

A potential divider has R₁ = 4.0 kΩ and R₂ = 6.0 kΩ in series across a 10 V supply. Calculate V_out across R₂.

6. Potential Dividers & Variable Resistance

The potentiometer

A potentiometer is a single component that acts as a whole potential divider by itself. It's a coil of resistance wire with a sliding contact that can move along it — imagine a volume knob. As the slider moves, it splits the coil into two parts with different resistances, so the output voltage taken from the slider changes continuously.

Circuit symbol: a rectangle (resistor) with an arrow pointing into it, representing the sliding contact.
If total resistance is, say, 3 Ω, the output can be varied smoothly between a minimum (0 Ω portion → V_out = 0) and a maximum (full 3 Ω portion → V_out = V_in).
Using only two of the three terminals turns a potentiometer into a simple variable resistor instead.

Sensory potential dividers: LDRs and thermistors

Swap one of the fixed resistors in a potential divider for a sensory resistor (an LDR or thermistor) and you get a circuit whose output voltage automatically responds to light or temperature — perfect for switching things on or off.

Component	Resistance responds to	How resistance changes
LDR (Light Dependent Resistor)	Light intensity	Higher light intensity → lower resistance. Lower light intensity → higher resistance.
Thermistor	Temperature	Hotter → lower resistance. Cooler → higher resistance.

Since V = IR (and current is the same throughout a series circuit), V_out across a sensory resistor increases as its resistance increases. And since V₁ + V₂ must always equal V_in, whatever happens to one resistor's share happens in reverse to the other's.

LDR circuit — street lights

As it gets dark, light intensity falls → LDR resistance increases → V_out across the LDR increases → this rising voltage can trigger a lamp to switch ON. That's how automatic street/security lights work.

Thermistor circuit — fire alarms / heaters

As temperature falls, thermistor resistance increases → V_out across the thermistor increases → this can trigger a heater to switch ON. As temperature rises (e.g. a fire), the opposite pattern can trigger an alarm.

Worked Example

A potential divider has a fixed resistor R in series with a thermistor. What happens to the p.d. across R and across the thermistor when the thermistor's temperature decreases?

Step 1 — Ohm's Law: As temperature decreases, thermistor resistance increases. Since current I is the same through both components (series circuit), p.d. across the thermistor (V=IR) must increase.

Step 2 — Voltages rule: The two p.d.s must always add up to V_in. If the thermistor's p.d. increases, the p.d. across R must decrease to compensate.

Answer: p.d. across thermistor increases; p.d. across R decreases.

Practice Question 1

An LDR is connected in series with a fixed resistor R, with V_out taken across the LDR. As the room gets brighter, what happens to V_out? Explain your reasoning fully.

Practice Question 2

Explain how a potentiometer can act as a variable resistor rather than a potential divider.

What to Memorise

Resistance

Opposition to current flow, caused by collisions between free electrons and fixed ions. Measured in ohms (Ω).

Resistivity (ρ)

A material property (independent of shape) describing how strongly it opposes current. Units: Ω m. Depends on temperature.

R = ρL/A

Resistance = resistivity × length ÷ cross-sectional area.

Drift velocity

The average velocity of charge carriers travelling through a conductor — typically ~10⁻³ m s⁻¹, much slower than you'd expect.

I = nqvA

Current = number density × charge × drift velocity × cross-sectional area.

Potential difference

Energy transferred (work done) per unit charge. V = W/Q. Measured in volts with a voltmeter connected in parallel.

V = IR

Ohm's Law — links potential difference, current, and resistance.

Potential divider equation

V_out = [R₂/(R₁+R₂)] × V_in — output taken across R₂.

V₁/V₂ = R₁/R₂

Ratio of p.d.s across series resistors equals the ratio of their resistances.

LDR

Resistance falls as light intensity rises. Used in light-sensing circuits (street lights).

Thermistor

Resistance falls as temperature rises. Used in temperature-sensing circuits (fire alarms, heaters).

Potentiometer

A variable resistor with a sliding contact; used as a full potential divider (3 terminals) or a simple variable resistor (2 terminals).

Concepts Checklist

I can explain why resistance occurs at a microscopic (electron-ion collision) level
I can use R = ρL/A and rearrange it for any variable
I understand why doubling diameter quarters resistance (area ∝ r²)
I can describe the full method for Core Practical 7, including control variables
I can identify systematic vs random errors in the resistivity experiment and how to reduce them
I can find resistivity from the gradient of an R vs L graph (ρ = gradient × A)
I understand drift velocity and why it's so slow despite current appearing instant
I can use I = nqvA and convert units correctly (mm² → m² etc.)
I know why insulators, conductors, and semiconductors have such different resistivities (in terms of n)
I can define potential difference as energy/work done per unit charge
I know a voltmeter is always connected in parallel
I can apply the potential divider equation and identify which resistor V_out is across
I can explain how increasing one resistor's value affects the p.d. across both resistors
I can explain how a potentiometer works as a variable potential divider
I can predict how V_out changes in LDR and thermistor circuits as conditions change

Exam Tips & Common Mistakes

Don't compare resistivity values directly to answer "which is the better conductor." Resistivity is a material property, but resistance also depends on shape (R=ρL/A). A material with higher ρ but a much bigger cross-sectional area can still have lower overall resistance — always calculate R for a fair comparison.

Always convert units to SI before substituting into formulas. Diameters/areas are usually given in mm or mm² — convert to metres/m² before using R=ρL/A or I=nqvA. This is one of the most common lost-marks reasons in this topic.

Check which resistor V_out is measured across before using the potential divider equation. The formula V_out=[R₂/(R₁+R₂)]×V_in assumes V_out is across R₂ — if the diagram shows it across the other resistor, swap the numerator.

Don't confuse ammeter and voltmeter placement. Ammeters go in series (measure current flowing through), voltmeters go in parallel (measure p.d. across). Mixing this up in a practical-based question is an easy way to lose marks.

Remember why low current matters in Core Practical 7. Examiners often ask "why keep the current small" — the answer is always about preventing resistive heating from changing the wire's resistivity mid-experiment, not just "to be safe."

Reasoning chains for sensor circuits (LDR/thermistor) should always be two steps: (1) how the physical change affects resistance, then (2) how that resistance change affects V_out via V=IR and the fact that p.d.s must sum to V_in. Markschemes reward this full chain of logic, not just the final answer.

Semiconductors behave oppositely to metals with temperature. A metal's resistance with temperature (more ion vibration → more collisions), but a semiconductor's resistance with temperature (more free electrons released). Don't mix these up.

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Also in the full note

3. Calculating Current & Drift Velocity
4. Potential Difference & Conductor Length
6. Potential Dividers & Variable Resistance
Exam Tips & Common Mistakes
Equipment & resolution

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