S4.12.3 Chapter 3

Ohm's law

Priya joins a resistor to a 3 V battery and measures the current. Then she doubles the p.d. to 6 V. What happens to the current?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A The current stays the same, because the resistor sets it.
You picked A. Read on. At the end, the chapter comes back to your guess.
B The current doubles.
You picked B. Read on. At the end, the chapter comes back to your guess.
C The current goes up by 3 A.
You picked C. Read on. At the end, the chapter comes back to your guess.
D The current halves.
You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

Use V equals I times R in calculations
Plot a graph of current against voltage for a resistor

A bigger p.d. pushes a bigger current through the same part. How much bigger? The answer has a number in it, and it was found by measuring. This chapter shows how the three quantities fit together: p.d., current and resistance.

The resistance of a part tells how hard it makes it for charge to flow. It is the p.d. across the part divided by the current through it: R = V ÷ I. It is measured in ohms, written Ω. A part has a resistance of 1 Ω when a p.d. of 1 V drives a current of 1 A through it. So 1 Ω = 1 V/A.

A resistor is a part made to have a certain resistance. A variable resistor has a resistance you can change, with a sliding contact. To see how current depends on p.d., set up a circuit that lets you change the p.d. across one resistor.

The symbols for the two kinds of resistor.

The test circuit. The ammeter reads the current through the resistor labelled R. The voltmeter, with the reading 3 V under it, is across that resistor. The variable resistor changes both.

Slide the variable resistor to a new place, then write down the voltmeter and the ammeter readings. Do this for six settings. Here are the readings for a resistor called R.

Six readings. Look at the last column. The p.d. divided by the current is the same every time.

Double the p.d. and the current doubles. Treble it and the current trebles. That pattern is called direct proportion. Plot the readings on a graph with the p.d. along the bottom and the current up the side.

Draw it first

The graph of current against p.d.

1 Plot each reading as a dot: p.d. along, current up.
2 The dots lie on a straight line, and the line goes through the origin. No p.d. means no current.
3 Read a point on the line, such as 4 V and 0.4 A. The p.d. divided by the current is 4 ÷ 0.4 = 10 Ω, as in the table.

A straight line through the origin means the current is proportional to the p.d. This is Ohm's law. For a part like this, current is directly proportional to p.d., as long as its temperature stays the same. A part that follows it is an ohmic conductor. A metal wire or a resistor at a steady temperature is one.

Because V ÷ I stays the same, we can write V = IR. Look closely at what this says. R = V ÷ I is the meaning of resistance, and it holds for every part. Ohm's law is the extra fact that R does not change for an ohmic conductor. For other parts, such as a lamp, V = IR still holds at each moment, but R changes.

See it three ways

As a picture

In words

The p.d. is the current times the resistance. The current is the p.d. divided by the resistance. The resistance is the p.d. divided by the current.

In symbols

V = IR, so I = V ÷ R and R = V ÷ I. The units agree: V = A × Ω, A = V ÷ Ω and Ω = V ÷ A.

You do not need to remember three equations. Start from V = IR and do the same to both sides. To get I alone, divide both sides by R: V ÷ R = I. To get R alone, divide both sides by I: V ÷ I = R. The letter you want ends up alone on one side.

Worked example

A resistor of 20 Ω carries a current of 0.3 A. What is the p.d. across it?

Write the equation: V = IR.
Put in the numbers: V = 0.3 A × 20 Ω.
V = 6 V.

Answer: 6 V

Worked example

A 12 V battery is joined across a resistor of 40 Ω. What current flows? Give the answer in milliamps.

We want I. Turn V = IR round to I = V ÷ R.
Put in the numbers: I = 12 V ÷ 40 Ω = 0.3 A.
Change to milliamps: 0.3 A × 1 000 = 300 mA.

Answer: 300 mA

Worked example

A lamp has a p.d. of 4.5 V across it and a current of 150 mA through it. What is its resistance at that moment?

Change the current to amps: 150 mA = 150 ÷ 1 000 = 0.15 A.
We want R. Turn V = IR round to R = V ÷ I.
Put in the numbers: R = 4.5 V ÷ 0.15 A = 30 Ω.
Check: 0.15 A × 30 Ω = 4.5 V, which is the p.d. we started with.

Answer: 30 Ω

Now compare different resistors on one graph. A bigger resistance lets less current through at the same p.d. So its line is lower, and shallower. The gradient of an I-V graph is I ÷ V, which is 1 ÷ R. A steep line means a small resistance.

Three resistors. The steepest line is the smallest resistance. The gradient of the line is 1 ÷ R.

Try it

A longer pencil, a bigger resistance

You need: two torch cells of 1.5 V each, a small torch bulb, two pencils, one short and one long, soft ones marked B if you have them, three pieces of bare wire or kitchen foil strips, sticky tape.

Be safe: Use only torch cells. The lead of a pencil is safe to touch. Switch off by lifting a wire when you have looked, because the lead can get warm. Use a pencil sharpener, never a knife.

Sharpen both ends of each pencil with a sharpener, so that the dark lead shows at both points. Do not use a knife.
Put the two cells end to end in a line, and tape a wire to each free end. Tape one cell wire to the tip at the bottom of the bulb. Tape a third wire to the bulb's metal side. This leaves one gap, between the third wire and the other cell wire.
Close the gap with the short pencil. Press the free end of the cell wire firmly on one point. Press the free end of the third wire on the other. Note how bright the bulb is.
Do the same with the long pencil, and compare the brightness.

What you will see: The long pencil gives the dimmer bulb. The graphite lead inside is longer, so it has a larger resistance. The p.d. of the cells stays the same, so V = IR gives a smaller current. The bulb gets less current and shines less. A soft, dark pencil works best.

If you cannot try it, this is what happens.

A wrong idea many people have

The resistance of a resistor gets bigger when you increase the p.d. across it.

Why it feels right: A bigger p.d. gives a bigger current, and a bigger current seems to mean more trouble for the charge.

The test that breaks it: Look at the table. The p.d. goes from 1 V to 6 V, and the last column stays at 10 Ω all the way down.

What is true: For an ohmic conductor at a steady temperature, the resistance does not change. A bigger p.d. gives a bigger current, but V ÷ I stays the same.

A wrong idea many people have

R = I ÷ V, or R = V × I.

Why it feels right: The three letters V, I and R can be put together in many ways. People often pick the wrong one.

The test that breaks it: Check the units. Ω is V per A, so R must be V ÷ I. For 4 V and 0.4 A, V × I gives 1.6 and I ÷ V gives 0.1. Neither is the 10 Ω of the graph.

What is true: Start from V = IR, and divide both sides by what you want to remove. R = V ÷ I, and I = V ÷ R. Check that the units of your answer agree.

A wrong idea many people have

The resistance is the gradient of the graph of current against p.d.

Why it feels right: The gradient of a graph is often the quantity of interest, and a steeper line looks like a bigger something.

The test that breaks it: Look at the graph of the three resistors. The steepest line belongs to the smallest resistance, 5 Ω. The gradient of an I-V line is 1 ÷ R.

What is true: On a graph of current against p.d., the gradient is 1 ÷ R. A steeper line means a smaller resistance. To find R, divide a p.d. by its current.

Key words

resistance: how hard a part makes it for charge to flow. R = V ÷ I.
ohm: the unit of resistance, written Ω. One ohm is one volt for each amp.
resistor: a part made to have a certain resistance.
variable resistor: a resistor whose resistance can be changed with a sliding contact.
Ohm's law: the current through an ohmic conductor is directly proportional to the p.d. across it, at a steady temperature.
ohmic conductor: a part that follows Ohm's law, so its resistance does not change.

Back to your guess

The answer is B: The current doubles.

For a resistor at a steady temperature, the current is proportional to the p.d. Double the p.d. and the current doubles. The resistance, which is the p.d. divided by the current, does not change.

Keep it

Copy this into your exercise book.

R = V ÷ I, in ohms. V = IR, I = V ÷ R. An ohmic conductor at a steady temperature gives a straight I-V line through the origin. Steeper means smaller R.

Draw: Axes with p.d. along and current up. Draw two straight lines through the origin, a steep one and a shallow one. Label the steep one small R and the other big R.