CDS Physics · Electricity and Magnetism

Resistance and Circuits

Ohm's law, how a wire's resistance depends on its length and thickness, resistivity as a property of the material, and resistors joined in series and in parallel.

All Electricity and Magnetism notes

Why this matters

Twelve CDS questions from 2017 (II) to 2026 (II), eleven of them MODERATE. Six are about resistivity (its unit, what changes it, which material has more), four join resistors in series or parallel, and two scale a wire's resistance with its length or diameter.

Concept 1 of 3: Ohm's law and the resistance of a wire

At a steady temperature, the current through a metal wire grows in step with the voltage across it: double the voltage, double the current. The ratio is the resistance. A longer wire is a longer road for the charge, so it resists more. A thicker wire is a wider road, so it resists less. Thickness counts through the area, which goes with the square of the diameter.

Definition

The rules:

Ohm's law: $V = IR$ . One ohm is one volt per ampere.
$R = \rho\dfrac{l}{A}$ : resistance goes up with length and down with area.
Area $A = \pi r^{2} = \dfrac{\pi d^{2}}{4}$ , so doubling the diameter makes the area 4 times and the resistance one quarter.
Stretching a wire to $n$ times its length keeps its volume, so its area becomes $1/n$ times and its resistance $n^{2}$ times.
With the same cell, $I = V/R$ : halve the resistance and the current doubles.

Ohm's law and resistance of a wire

V = IR \qquad R = \rho\dfrac{l}{A} \qquad A = \pi r^{2} = \dfrac{\pi d^{2}}{4}

ρresistivity of the material
llength of the wire
Aarea of cross-section
r, dradius and diameter of the wire

Worked example

A copper wire has a resistance of 6 Ω. A second copper wire is 3 times as long and has 3 times the diameter. Find its resistance, and the current each wire draws from a 12 V cell.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (II) 2022 — General Knowledge · Q28Moderate

Example 1 · Electricity and Magnetism · Resistance and Circuits

A metal wire of length

l

and diameter

d

has a resistance

R

. What would be the resistance of another wire of the same metal and of same length but having double the diameter ?

Thickness counts through the square of the diameter

Resistance depends on the area, and area goes as

d^{2}

. Doubling the diameter divides the resistance by 4, not by 2.

Stretching changes both length and area

Pull a wire to

n

times its length and it also gets thinner. Its resistance becomes

n^{2}

times, not

n

times.

Drill 1 more on ohm's law and the resistance of a wire

Concept 2 of 3: Resistivity: a property of the material

Resistance belongs to one particular wire. Resistivity belongs to the material it is made of. A long thin copper wire and a short thick one have different resistances, but the same resistivity, because both are copper. Only the material and its temperature change resistivity.

Definition

What to know:

$\rho = \dfrac{RA}{l}$ . Its unit is the ohm metre (Ω m), or the ohm centimetre.
It depends on the material and the temperature only, never on the length or area of the wire.
Metals: about $10^{-8}$ Ω m. Insulators such as glass and rubber: $10^{10}$ Ω m and more.
For a metal, resistivity rises as it gets hotter.
Alloys (nichrome, manganin, constantan) have a much higher resistivity than pure metals and do not oxidise easily when hot, so they are used in heaters, irons and toasters.
Same size wires, same current: the one of higher resistivity has more resistance and so gives out more heat ( $I^{2}R$ ).

Material	Resistivity at 20 °C (Ω m)	Where it is used
Silver	1.60 × 10⁻⁸	The best conductor; too costly for wiring
Copper	1.62 × 10⁻⁸	House wiring
Aluminium	2.63 × 10⁻⁸	Overhead power lines (light and cheap)Q CDS 2017 (II) and 2018 (I): an aluminium wire of the same size as a copper wire has more resistance, so it gives out more heat for the same current.
Tungsten	5.20 × 10⁻⁸	Bulb filaments
Nichrome (alloy)	About 1.1 × 10⁻⁶	Heater coils
Glass	10¹⁰ to 10¹⁴	An insulatorQ CDS 2022 (I): silver, nichrome and glass in increasing order of resistivity.
Rubber	10¹³ to 10¹⁶	Covering on wires

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (I) 2021 — General Knowledge · Q103Moderate

Example 2 · Electricity and Magnetism · Resistance and Circuits

We are given three copper wires of different lengths and different areas of cross-section. Which one of the following would have highest resistivity ?

Resistivity is not resistance

Changing the length or area of a wire changes its resistance, never its resistivity. Resistivity changes only with the material and the temperature.

The unit is ohm metre, not ohm per metre

From

\rho = RA/l

: ohm × m² ÷ m = ohm metre (or ohm centimetre). Ohm per metre and ohm per square metre are wrong.

Aluminium resists more than copper

Copper's resistivity is lower than aluminium's. So for wires of the same size, the aluminium one has the larger resistance and gives out more heat for the same current.

Drill 5 more on resistivity: a property of the material

Concept 3 of 3: Resistors in series and in parallel

In series the current has one path, so it must push through every resistor in turn and the resistances add. In parallel the current has several paths side by side, so it gets through more easily and the total is less than the smallest one. Across a fixed battery, less resistance means more current and so more heat every second.

Definition

The rules:

Series: $R = R_1 + R_2 + R_3$ . The same current flows through each; the voltages add. The total is more than the largest.
Parallel: $\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3}$ . The same voltage is across each; the currents add. The total is less than the smallest.
$n$ equal resistors $R$ : series $nR$ , parallel $R/n$ . Two in parallel: $\dfrac{R_1R_2}{R_1 + R_2}$ .
Heat per second across a fixed voltage is $V^{2}/R$ , so the all-parallel joining gives the most heat and the all-series the least.
Home appliances are joined in parallel: each gets the full mains voltage and can be switched on its own.
In a steady DC circuit a charged capacitor passes no current, so its branch carries none.

Series and parallel

R_s = R_1 + R_2 + R_3 \qquad \dfrac{1}{R_p} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3}

Rₛtotal resistance in series
Rₚtotal resistance in parallel

Worked example

Resistors of 4 Ω, 6 Ω and 12 Ω are joined to a 12 V battery, first all in parallel and then all in series. Find the total resistance, the current from the battery and the heat produced each second in each case.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2023 · CDS (I) 2023 — General Knowledge · Q44Moderate

Example 3 · Electricity and Magnetism · Resistance and Circuits

Three resistors of resistances

11\ \Omega

22\ \Omega

and

33\ \Omega

are connected in parallel. Their equivalent resistance is equal to

The parallel total is smaller than the smallest resistor

Adding a resistor in parallel always lowers the total, because it opens another path. If your parallel answer is bigger than the smallest resistor, the working has gone wrong.

Most heat from a battery comes from the smallest total resistance

Across a fixed voltage the heat per second is

V^{2}/R

. The smallest

R

, the all-parallel joining, gives the most heat. The rule

I^{2}R

(bigger

R

, more heat) holds only when the current is fixed.

Drill 3 more on resistors in series and in parallel

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Ohm's law and the resistance of a wire
Ohm's law and resistance of a wire
$V = IR \qquad R = \rho\dfrac{l}{A} \qquad A = \pi r^{2} = \dfrac{\pi d^{2}}{4}$
Resistors in series and in parallel
Series and parallel
$R_s = R_1 + R_2 + R_3 \qquad \dfrac{1}{R_p} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3}$

Reference tables (1)

Resistivity: a property of the material7 rows

Material	Resistivity at 20 °C (Ω m)	Where it is used
Silver	1.60 × 10⁻⁸	The best conductor; too costly for wiring
Copper	1.62 × 10⁻⁸	House wiring
Aluminium	2.63 × 10⁻⁸	Overhead power lines (light and cheap)Q CDS 2017 (II) and 2018 (I): an aluminium wire of the same size as a copper wire has more resistance, so it gives out more heat for the same current.
Tungsten	5.20 × 10⁻⁸	Bulb filaments
Nichrome (alloy)	About 1.1 × 10⁻⁶	Heater coils
Glass	10¹⁰ to 10¹⁴	An insulatorQ CDS 2022 (I): silver, nichrome and glass in increasing order of resistivity.
Rubber	10¹³ to 10¹⁶	Covering on wires

Watch out for (7)

Thickness counts through the square of the diameter
→ Ohm's law and the resistance of a wire
Stretching changes both length and area
→ Ohm's law and the resistance of a wire
Resistivity is not resistance
→ Resistivity: a property of the material
The unit is ohm metre, not ohm per metre
→ Resistivity: a property of the material
Aluminium resists more than copper
→ Resistivity: a property of the material
The parallel total is smaller than the smallest resistor
→ Resistors in series and in parallel
Most heat from a battery comes from the smallest total resistance
→ Resistors in series and in parallel

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