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Electricity and magnetism · Subtopic 3 of 8

Resistance

Resistance tells you how hard it is for current to flow: $R=\dfrac VI$ , in ohms (Ω). A fixed resistor at constant temperature has a straight-line $I$ – $V$ graph; a filament lamp curves because its resistance rises as it heats; a diode lets current through in one direction only. Longer, thinner wires have more resistance.

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Section 1 of 14

Learning objectives

Use $R=\dfrac VI$ , rearranged as needed.
Describe how to measure resistance with an ammeter and voltmeter.
Sketch and explain $I$ – $V$ graphs for a resistor, filament lamp and diode.
Use $R\propto L$ and $R\propto\dfrac1A$ for wires.

Pause and think

The same battery is connected first to a short, thick copper wire, then to a long, thin one. Which carries more current? Why?

Section 2 of 14

Core idea

Start hereWhy a lamp glows but the wires don't

The same current passes through the thin filament and the thick copper leads, yet only the filament glows. The filament resists the current far more, so that's where the energy is transferred.

The big idea

$R=\frac VI$ (Ω). Ohmic conductor: $I\propto V$ . $R\propto$ length, $R\propto\frac1{\text{area}}$ .

1. I–V graphs

Current–voltage graphs

Resistor: straight line through the origin, constant

R

. Lamp: curves over, because a hotter filament resists more. Diode: almost no current backwards.

V=IR

$R$: resistance (Ω = V/A).

Ohmic resistor: straight line through the origin.
Filament lamp: curve flattens; resistance rises as it heats.
Diode: current one way only.

Worked exampleAn ohmic resistor

6 V gives 0.3 A.

Goal: $R$ ; current at 9 V.
Strategy: $R=V/I$ ; constant $R$ .

$R=\frac6{0.3}=20\ \Omega$
Definition.
$I=\frac9{20}=0.45\text{ A}$
Same $R$ .

Answer

20\ \Omega

;

0.45

✓ Check

V

×1.5,

I

×1.5 ✓.

2. Wire dimensions

Resistance of a wire is proportional to its length

Measured points scatter slightly about a straight line through the origin.

Worked exampleA 4 Ω wire

Double the length; separately, double the diameter.

Goal: New resistances.
Strategy: $R\propto\frac LA$ ; area $\propto d^2$ .

$2\times4=8\ \Omega$
Longer path.
$A\times4\Rightarrow R=\frac44=1\ \Omega$
Double diameter: four times the area.

Answer

8\ \Omega

;

1\ \Omega

✓ CheckThick wires for high currents ✓.

Common misconceptionDiameter vs area Doubling the diameter quarters the resistance, not halves it.

Pause and think

Why does a thermistor's resistance fall when hot?

Summary

$V=IR$ .
$R\propto L$ , $R\propto1/A$ .

Ohm: V/A.

HistoryOhm's rejected discovery. Georg Ohm, a schoolteacher, published his law that current is proportional to voltage in

1827

. Critics called it "a web of naked fancies" and he lost his job. Fifteen years later the Royal Society gave him its top medal, and the unit of resistance is named after him.

Fun factSuperconductors: zero resistance. Some materials, cooled to extremely low temperatures, lose all resistance. A current started in a superconducting loop has kept flowing for years with no battery. MRI scanners use superconducting coils to make their strong magnetic fields.

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