D. Physics: Principles and Applications of Electricity · Electricity: Charges, Circuits and Energy

Investigating Current, Potential Difference and Resistance: Ohm's Law

D2.4 A1.2 A1.4 A1.5About 75 minutes

Learning goals

I can plan and carry out an investigation of how current depends on potential difference and on resistance.
I can graph potential difference against current and find the resistance from the slope.
I can state Ohm's law, V = I × R, as a mathematical model of the relationship.
I can tell ohmic from non-ohmic components using a graph.
I can use a short program to process investigation data.

Words to know

Ohm's law
For many conductors at constant temperature, the potential difference across them is directly proportional to the current through them: V = I × R.

directly proportional
Two quantities are directly proportional when doubling one doubles the other. The graph is a straight line through the origin.

inversely proportional
When one quantity doubles, the other halves. For a fixed voltage, current is inversely proportional to resistance.

slope
The steepness of a line on a graph: rise ÷ run. On a V–I graph, the slope is the resistance.

line of best fit
A straight line (or smooth curve) drawn to pass as close as possible to all the data points, showing the trend.

ohmic
Describes a component whose resistance stays constant, so it obeys Ohm's law (straight-line V–I graph).

non-ohmic
Describes a component whose resistance changes as the current changes, such as a filament lamp or an LED.

A question worth investigating

In 1827, Georg Ohm published the results of careful experiments with wires and electric current. He wanted to know: if you push harder (increase the potential difference), how much more current flows? Today you will repeat his investigation, find the pattern in the data, and build the mathematical model that every electrician and engineer uses.

The investigation

Question: How does the current through a resistor depend on the potential difference across it?

Hypothesis: If the potential difference across a resistor increases, then the current through it will increase in proportion, because a bigger push gives each charge more energy to move through the same resistance.

Variables:

Independent: potential difference (V), changed by adding cells or turning a supply dial.
Dependent: current (I), read on the ammeter.
Controlled: the same resistor, the same wires, and a constant temperature (take each reading quickly and open the switch between readings so the resistor does not heat up).

The set-up. The ammeter is in series with the resistor R; the voltmeter is connected across R. The potential difference is changed in steps (by adding cells or turning the dial of a variable DC power supply), and the current is recorded each time.

Potential difference V (V)	Current I (mA)	Current I (A)	V ÷ I (Ω)
0	0	0	–
1.5	16	0.016	94
3.0	29	0.029	103
4.5	46	0.046	98
6.0	59	0.059	102

Sample data for one resistor. V ÷ I is almost the same each time; the small differences come from measurement uncertainty.

Finding the pattern

When V doubles from 3.0 V to 6.0 V, the current about doubles, from 29 mA to 59 mA. The ratio V ÷ I stays about the same: close to 100 Ω.

Plot V on the vertical axis and I on the horizontal axis. The points lie close to a straight line through the origin. That means V is directly proportional to I. The slope of the line is:

slope = rise ÷ run = ΔV ÷ ΔI = R

The sample data graphed. The line of best fit passes through the origin and close to every point. Its slope, 6.0 V ÷ 0.060 A, is 100 Ω: the resistance of the resistor.

The mathematical model: Ohm's law

The straight line means V = (constant) × I, and the constant is the resistance. So:

V = I × R which can also be written I = V ÷ R or R = V ÷ I

This is Ohm's law. It tells you two relationships at once:

For a fixed resistance, current is directly proportional to potential difference (double V → double I).
For a fixed potential difference, current is inversely proportional to resistance (double R → half the current).

Resistance R (Ω)	50	100	200	300
Current I at 6.0 V (A)	0.12	0.060	0.030	0.020

Part 2 of the investigation: keep V = 6.0 V and change the resistor. Doubling R halves I; tripling R makes I one-third.

Worked example

Example 1: Use the model to predict

Using the resistor from the sample data (R = 100 Ω), predict the current when the potential difference is 9.0 V.

I = V ÷ R
I = 9.0 V ÷ 100 Ω = 0.090 A
Convert to mA: 0.090 × 1000 = 90 mA.

I = 0.090 A (90 mA). On the graph, this point would lie on the same straight line, extended.

Worked example

Example 2: Slope from a different graph

A student's V–I graph for a different resistor is a straight line through the origin and the point (0.25 A, 5.0 V). What is the resistance?

Slope = rise ÷ run = ΔV ÷ ΔI.
= (5.0 V − 0 V) ÷ (0.25 A − 0 A)
= 20 V/A

R = 20 Ω

Ohmic and non-ohmic components

A resistor at constant temperature gives a straight-line V–I graph: it is ohmic. Some components do not. A filament lamp gets hotter as the current increases, and hotter metal has more resistance, so its graph curves and gets steeper. LEDs are also non-ohmic. Ohm's law (V = I × R) still tells you the resistance at that moment, but R is not constant.

V–I graphs. A resistor gives a straight line (constant R). A filament lamp's graph curves upward because its resistance increases as it heats up.

V = [1.5, 3.0, 4.5, 6.0] # potential difference (V)

I = [0.016, 0.029, 0.046, 0.059] # current (A)

total = 0

for v, i in zip(V, I):

R = v / i

total = total + R

print(v, "V ->", round(R), "ohms")

print("Mean R =", round(total / len(V)), "ohms")

Python code (A1.4) that calculates R = V ÷ I for each reading and the mean. Output: 1.5 V -> 94 ohms, 3.0 V -> 103 ohms, 4.5 V -> 98 ohms, 6.0 V -> 102 ohms, Mean R = 99 ohms.

Graphing checklist

Independent variable usually goes on the x-axis. Here we put V on the y-axis on purpose, so that slope = R.
Title, axis labels with units, even scales.
Draw one line of best fit (do not join the dots).
Pick two points on the line, far apart, to find the slope.

Hands-on activity

Virtual (or real) Ohm's law lab

You need: a device with internet, or a real kit with a battery holder for 1 to 4 AA cells, a 100 Ω resistor, and a multimeter.

Virtual option: open the free PhET simulation "Circuit Construction Kit: DC". Build the circuit in the diagram above: battery, resistor, ammeter in series, voltmeter across the resistor.

Set the resistor to 10 Ω. Change the battery voltage in 5 steps (for example 2, 4, 6, 8, 10 V). Record V and I in a table.
Graph V (y-axis) against I (x-axis), draw the line of best fit, and find the slope.
Keep the voltage at 10 V and change the resistance (5, 10, 20, 40 Ω). Record I each time.
Write your conclusion: is your hypothesis supported? What mathematical model fits?

Real option: use 1, 2, 3 then 4 AA cells (1.5 V, 3.0 V, 4.5 V, 6.0 V) with the 100 Ω resistor, and measure V and I with the multimeter. Open the circuit between readings.

What to look for: in the 10 Ω simulation, 2 V gives 0.2 A, 4 V gives 0.4 A, and so on: a straight line with slope 10 Ω. At 10 V, 5 Ω gives 2 A, 10 Ω gives 1 A, 20 Ω gives 0.5 A, 40 Ω gives 0.25 A: doubling R halves I.

Safety (grown-up note): Use only AA batteries, never wall power. Never connect a wire or an ammeter straight across a battery. Keep cells away from young children.

Just for fun

Where does a resistor feel most relaxed after a long day of opposing current?

Ohm, sweet Ohm.

Your turn

State Ohm's law in words and as an equation.

What does the slope of a V–I graph (V on the y-axis) represent?

A resistor has 2.0 V across it and 0.10 A through it. If V is increased to 6.0 V, what current do you predict?

At a fixed 12 V, the resistance in a circuit is doubled. What happens to the current?

Why should the switch be opened between readings?

A V–I graph is a curve that gets steeper. Is the component ohmic? What is happening to its resistance?

More practice: warm-up

What is the dependent variable in the first part of the investigation?

Which meter is connected across the resistor?

Write Ohm's law solved for I.

Is a filament lamp ohmic or non-ohmic?

If V doubles for an ohmic resistor, what happens to I?

More practice: on your own

A V–I graph line passes through (0, 0) and (40 mA, 2.0 V). Find the resistance.

Data: 1.0 V, 0.20 A; 2.0 V, 0.41 A; 3.0 V, 0.59 A; 4.0 V, 0.80 A. Is the component ohmic? Estimate R.

Two resistors are tested. On the same V–I axes, resistor A's line is steeper than B's. Which has more resistance?

Explain why the line of best fit should go through the origin.

Predict the output of this code: V = 12 then for R in [4, 6, 12]: print(V / R)

Challenge

A student's data for a resistor: 2.0 V, 0.10 A; 4.0 V, 0.20 A; 6.0 V, 0.36 A; 8.0 V, 0.40 A. Which reading looks wrong, and how should the student deal with it?

Write a short Python program that prints the current through a 220 Ω resistor for potential differences of 1.1 V, 2.2 V and 4.4 V. Show the output.

A filament lamp has 1.0 V across it at 0.20 A, and 6.0 V across it at 0.50 A. Calculate its resistance at each point and explain the difference.

Check your understanding

Quiz

Ohm's law is written as:

On a graph with V on the y-axis and I on the x-axis, the slope equals:

For an ohmic resistor, the potential difference is tripled. The current:

At a fixed potential difference, which change halves the current?

Why is a filament lamp non-ohmic?

A straight-line V–I graph passes through (0.30 A, 6.0 V). What is R?

For parents and teachers

Big idea: Expectation D2.4 asks students to investigate the relationships among current, potential difference and resistance and develop a mathematical model: that model is Ohm's law, V = I × R. Students also practise the experimentation process (A1.2) and coding (A1.4).

Common difficulties:

Slope. Students need rise ÷ run with units: (V) ÷ (A) = Ω. Remind them to convert mA to A.
Direct vs inverse: at fixed R, V and I go up together; at fixed V, more R means less I.
Joining the dots instead of drawing a line of best fit.

How to help: The free PhET simulations ("Ohm's Law" and "Circuit Construction Kit: DC") let students collect real-looking data safely at home. If you use real batteries, use AA cells only and never connect a wire directly across a battery.