Ohm’s law

V = RI

Narrated video. Transcript:

In 1828, Georg Simon Ohm studied how the current in a conductor depends on the voltage across it. Here is his experiment: a wire, a variable supply, an ammeter in series and a voltmeter across the wire.

Raise the voltage step by step and note the current. 2 volt gives 1 ampere, 4 volt gives 2 ampere, 6 volt gives 3 ampere. Double the voltage, and the current doubles.

Plot V against I and the points lie on a straight line through the origin. So V is proportional to I: V equals R I. This is Ohm’s law.

The constant R is the resistance of the conductor. Its SI unit is the ohm, written with the Greek letter Ω. One ohm is one volt per ampere.

What does R depend on? Take a slab of material, of length l and cross-section A. A voltage V drives a current I through it.

Now put two identical slabs end to end. The same current I flows, but each slab needs V across it, so the total is 2V. The resistance has doubled. R is proportional to the length l.

Instead, put the two slabs side by side. Same voltage V across them, but each carries I, so the total current is 2I. The resistance has halved. R is inversely proportional to the area A.

Put both together: R equals ρ l over A. The constant ρ, rho, is the resistivity. It depends only on the material and its temperature, not on the shape. Its unit is the ohm metre.

Current density and resistivity

Narrated video. Transcript:

Let us write Ohm’s law in a way that does not depend on the size of the wire. The current per unit area, normal to the flow, is called the current density, j equals I over A. Its unit is ampere per square metre.

Inside a uniform wire of length l, the field is uniform, so the voltage across it is V equals E l.

Now put V equals E l and R equals ρ l over A into V equals R I. We get E l equals ρ l times I over A. The l cancels, and I over A is j. So E equals ρ j.

Writing σ, sigma, for 1 over ρ, this is j equals σ E. σ is the conductivity. This is Ohm’s law at a point inside the material: the current density is proportional to the field.

Worked example. A nichrome wire is 1 metre long, with a cross-section of 0.5 square millimetre, that is 0.5 × 10⁻⁶ square metre. The resistivity of nichrome is about 1.0 × 10⁻⁶ ohm metre.

R equals ρ l over A: 1.0 × 10⁻⁶ times 1, divided by 0.5 × 10⁻⁶. That gives 2 ohm.

Connect it to a 6 volt supply. I equals V over R, 6 over 2, so 3 ampere flows.

The current density is 3 divided by 0.5 × 10⁻⁶, which is 6 × 10⁶ ampere per square metre. The field is 6 volt over 1 metre, 6 volt per metre.

Check E equals ρ j: 1.0 × 10⁻⁶ times 6 × 10⁶ is 6 volt per metre. It agrees. A copper wire of the same size would have about sixty times less resistance.

Try it: Ohm’s experiment

Interactive simulation. Things to try:

• Slide the voltage from 0 to 12 V. Points appear on the graph, and they all lie on one straight line through the origin.
• Double the length: a new, less steep line appears. The resistance has doubled.
• Double the area instead: the line gets steeper. The resistance has halved.
• Compare the readouts for E = V/l and j = I/A: E/j always equals ρ, whatever the length or area.
• Switch to copper with the same size. Watch how small R becomes, and how big the current gets.

Stretch the wire

A wire of resistance 2 Ω is stretched uniformly to twice its original length. Its volume does not change. What is its new resistance?

1. 1 Ω
2. 2 Ω
3. 4 Ω
4. 8 Ω

Same material, bigger wire

Wire B is made of the same material as wire A, but is twice as long and has twice the cross-sectional area. Which statement is correct?

1. B has twice the resistance and twice the resistivity of A
2. B has the same resistance and the same resistivity as A
3. B has the same resistance but twice the resistivity of A
4. B has four times the resistance of A

Summary

• Ohm’s law: V = R I
• Resistance: R = ρ l / A
• Current density: j = I / A
• Ohm’s law at a point: E = ρ j, j = σ E, σ = 1/ρ

• For a conductor obeying Ohm’s law, V is proportional to I; the V–I graph is a straight line through the origin.
• Resistance (unit ohm, Ω) is proportional to the length and inversely proportional to the area of cross-section.
• Resistivity ρ (unit Ω m) depends only on the material and temperature, not on the size or shape.
• The conductivity σ = 1/ρ links the current density to the field: j = σE.