Resistance and resistivity: R = ρl/A

The idea

A uniform metallic conductor's resistance is directly proportional to its length l, inversely proportional to its cross-sectional area A, and depends on its material: R = ρl/A, where electrical resistivity ρ (SI unit Ω m) is characteristic of the material.

A long thin wire

In a circuit, the ammeter reading falls to one half when the length of a wire is doubled. It rises when a thicker wire of the same material and length is used.

A wire has some resistance. Which change would make its resistance larger?

• Making it longer, with the same material and thickness
• Making it shorter, with the same material and thickness
• Making it thicker, with the same material and length

Length, thickness, material

Change length, cross-section and material. Watch the orange bar for the resistance.

Three things

A longer wire resisted more. A thicker wire resisted less. A different material changed the resistance too. All three are packed in one formula.

A uniform metallic conductor's resistance is directly proportional to its length l, inversely proportional to its area of cross-section A, and depends on its material: R = ρl/A. Here ρ (rho), the electrical resistivity, is characteristic of the material. Its SI unit is Ω m.

Notes

A uniform conductor's resistance is directly proportional to its length and inversely proportional to its area of cross-section, and depends on its material: R = ρl/A, where resistivity ρ (unit Ω m) is characteristic of the material.

Check yourself

A wire of a given material has a resistance of 4 Ω. What is the resistance of another wire of the same material with half the length and twice the area of cross-section, in Ω?

Answer: 1 Ω

R is proportional to l/A. With l halved and A doubled, R becomes 1/2 × 1/2 = 1/4 of 4 Ω, which is 1 Ω.

The length of a uniform wire is doubled, with the same material and area. What happens to its resistance?

A thicker wire of the same material and the same length is used. What happens to the resistance?

Which quantity in R = ρl/A is characteristic of the material of the wire?

• It doubles — correct. Yes!
• It halves. Resistance is directly proportional to length, so doubling the length doubles the resistance.
• It does not change. The resistance depends on the length, so a longer wire has a larger resistance.
• It increases. Resistance is inversely proportional to the area of cross-section, so a thicker wire has less resistance.
• It decreases — correct. Yes!
• It stays the same. The area of cross-section changes the resistance. A larger area gives a smaller resistance.
• The length l. The length can be changed for a wire of the same material. The resistivity ρ belongs to the material.
• The area A. The area can be changed for the same material. The resistivity ρ belongs to the material.
• The resistivity ρ — correct. Yes!