SPI Physics · Electricity and Magnetism

Ohm's Law and Resistance

The current through a metal wire is proportional to the voltage across it, so V = IR, and the wire's resistance R depends on its length, its thickness and what it is made of.

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Why this matters

Five questions in the bank, three EASY and two MODERATE. Three are one line of V = IR, one reads a V against I graph, and one asks which metal resists current the most. The marks go on dividing the wrong way round and on forgetting that Ohm's law needs a steady temperature.

Concept 1 of 3: Current, charge and potential difference

An electric current is charge on the move, like water flowing in a pipe. The potential difference is the push that drives it: the work done to move each coulomb of charge from one point to the other.

Definition

Current $I = \dfrac{Q}{t}$ : charge passing a point each second. Unit: ampere (A). $1\text{ mA} = 0.001\text{ A}$ .
Potential difference $V = \dfrac{W}{Q}$ : work done per coulomb. Unit: volt (V).
Charge is measured in coulombs (C).
An ammeter goes in series with the part it measures; a voltmeter goes in parallel across it.

Current and potential difference

I = \frac{Q}{t} \qquad V = \frac{W}{Q}

Qcharge, in coulombs
ttime, in seconds
Wwork done, in joules

Worked example

A charge of 30 C flows through a lamp in one minute. Find the current.

Practice this conceptself-check · 4 quick reps

Time in minutes

Current is coulombs per SECOND. 30 C in one minute is

\frac{30}{60} = 0.5

A, not 30 A.

Concept 2 of 3: Ohm's law: V = IR

Double the push across a metal wire and twice as much current flows. The fixed ratio of voltage to current is the wire's resistance: how hard it is for current to get through.

Definition

Ohm's law: at constant temperature, the current through a metal conductor is proportional to the potential difference across it.
So $V = IR$ , and the same rule rearranged gives $I = \dfrac{V}{R}$ and $R = \dfrac{V}{I}$ .
Resistance is measured in ohms ( $\Omega$ ): $1\,\Omega = 1\text{ V per A}$ .
A graph of V against I is a straight line through the origin, and its slope is R. The ratio V/I is the same at every point.
If the temperature changes (a glowing filament), R changes and the graph bends: the law no longer holds.

Ohm's law

V = IR \qquad I = \frac{V}{R} \qquad R = \frac{V}{I}

Worked example

A current of 0.5 A flows through a 12

\Omega

resistor. Find the potential difference across it.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Electricity and Magnetism · Ohm's Law and Resistance

A resistance of 20 ohms has a current of 2 amperes flowing in it. What potential difference is there between its ends?

Dividing the wrong way round

Resistance is

R = \frac{V}{I}

. For 6 V and 4 A that is 1.5

\Omega

;

\frac{I}{V} = \frac{4}{6}

is not a resistance at all.

Thinking V/I changes along a straight-line graph

A straight line through the origin means V/I is the SAME at every point. That is what Ohm's law says, so the conductor is ohmic.

Leaving out the temperature condition

Ohm's law holds for a metal only while its temperature stays constant. A hot filament's resistance rises, so its V against I graph curves.

Drill 3 more on ohm's law: v = ir

Concept 3 of 3: What a wire's resistance depends on

A longer wire is a longer obstacle course, so it resists more. A thicker wire gives the current more room, so it resists less. And some metals let current through far more easily than others.

Definition

$R = \rho \dfrac{L}{A}$ : resistance is proportional to length and inversely proportional to cross-section area.
$\rho$ (rho) is the resistivity. It depends only on the material and the temperature, not on the wire's size. Unit: ohm metre ( $\Omega\,\text{m}$ ).
Best conductors (lowest resistivity): silver, then copper, then aluminium. Copper and aluminium carry current in house wiring.
Tungsten has a much higher resistivity and melts at about $3400^\circ\text{C}$ , so it makes bulb filaments.
Nichrome (an alloy) has a high resistivity and does not burn easily when red hot, so it makes heater elements.

Resistance of a wire

R = \rho \frac{L}{A}

ρresistivity of the material
Llength of the wire
Aarea of cross-section

Worked example

A copper wire 2 m long has a resistance of 4

\Omega

. What is the resistance of a 5 m wire of the same copper and the same thickness?

Practice this conceptself-check · 5 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Electricity and Magnetism · Ohm's Law and Resistance

Out of the following ____ is the most resistant metal.

Mixing resistance with resistivity

Cutting or stretching a wire changes its resistance, but its resistivity stays the same: resistivity belongs to the material, not the wire.

Thicker means more resistance

It is the other way round. Doubling the area of cross-section halves the resistance, because the current has more room.

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 (3)

Current, charge and potential difference
Current and potential difference
$I = \frac{Q}{t} \qquad V = \frac{W}{Q}$
Ohm's law: V = IR
Ohm's law
$V = IR \qquad I = \frac{V}{R} \qquad R = \frac{V}{I}$
What a wire's resistance depends on
Resistance of a wire
$R = \rho \frac{L}{A}$

Watch out for (6)

Time in minutes
→ Current, charge and potential difference
Dividing the wrong way round
→ Ohm's law: V = IR
Thinking V/I changes along a straight-line graph
→ Ohm's law: V = IR
Leaving out the temperature condition
→ Ohm's law: V = IR
Mixing resistance with resistivity
→ What a wire's resistance depends on
Thicker means more resistance
→ What a wire's resistance depends on

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