Aim for FivePhysics 2

Unit 11 · Topic 11.3

11.3 Resistance, Resistivity, and Ohm’s Law

Resistance measures how hard it is for current to pass through an element. A wire's resistance depends on its material, length and thickness through R = ρℓ/A, and for ohmic elements Ohm's law, I = ΔV/R, links current to potential difference with a constant R. Graphs of current against potential difference show which elements are ohmic.

Key terms

  • resistance
  • resistivity
  • ohm
  • Ohm's law
  • ohmic and nonohmic

Resistance and resistivity

Resistance R tells you how strongly an element opposes the flow of charge. Its unit is the ohm (Ω). For a piece of material with a uniform shape, like a wire:

R=ρℓAR = \frac{\rho\ell}{A}

Here ℓ is the length, A is the cross-sectional area, and ρ (rho) is the resistivity, measured in Ω·m. Longer wires have more resistance, since charge has farther to push through. Thicker wires have less, since there are more paths side by side.

Resistivity is a property of the material itself, set by its atomic structure. Copper's is about 1.7 × 10⁻⁸ Ω·m, while rubber's is enormous. For most conductors, resistivity rises as they get hotter, because the atoms vibrate more and get in the way of the drifting electrons.

Ohm's law

For many materials, the current through an element is proportional to the potential difference across it:

I=ΔVRI = \frac{\Delta V}{R}

Elements that follow this with a constant R, whatever the current, are called ohmic. On the AP exam, resistors and bulbs are treated as ohmic unless a question says otherwise. In this model, an ohmic material's resistivity is treated as constant even as it warms up.

Resistors turn electrical energy into thermal energy, which can raise the temperature of the resistor and its surroundings. That's how toasters and space heaters work.

Current–voltage graphs

Plotting current (vertical) against potential difference (horizontal) is the standard way to test an element.

  • Ohmic element: a straight line through the origin. The slope is 1/R, so a steeper line means less resistance.
  • Real lightbulb filament: the line starts straight but bends toward the potential difference axis at higher voltages, because the hot filament's resistance rises.
  • If the graph is drawn the other way, ΔV against I, the slope is R itself. Always check which variable is on which axis.

In the lab

To find a wire's resistivity, you might measure the resistance of different lengths of the same wire. A graph of R against ℓ is a straight line with slope ρ/A, so ρ = slope × A. Measure the diameter to find A = πr². Keep the current small so the wire doesn't heat up and change its resistance.

Worked examples

Try each one yourself first, then open the solution.

  1. Example 1Calculator allowed

    Resistance of a copper wire

    Find the resistance of a copper wire 10 m long with a diameter of 1.0 mm. Copper's resistivity is 1.7 × 10⁻⁸ Ω·m.

    Show the solution
    1. Step 1: Radius = 0.50 mm = 5.0 × 10⁻⁴ m. Area A = πr² = π(5.0 × 10⁻⁴)² ≈ 7.85 × 10⁻⁷ m².
    2. Step 2: R=ρℓA=(1.7×10−8)(10)7.85×10−7≈0.22R = \dfrac{\rho\ell}{A} = \dfrac{(1.7 \times 10^{-8})(10)}{7.85 \times 10^{-7}} \approx 0.22 Ω.

    Answer: About 0.22 Ω

  2. Example 2

    Stretching a wire (classic trap)

    A wire with resistance R is stretched so its length doubles. Its volume stays the same. What is its new resistance?

    Show the solution
    1. Step 1: The quick wrong answer is 2R, from doubling ℓ alone.
    2. Step 2: Stretching also makes the wire thinner. Volume = ℓA stays constant, so if ℓ doubles, A must halve.
    3. Step 3: Rnew=ρ(2ℓ)A/2=4ρℓA=4RR_{\text{new}} = \dfrac{\rho(2\ell)}{A/2} = 4\dfrac{\rho\ell}{A} = 4R.

    Answer: 4R

  3. Example 3Calculator allowed

    Resistance from a graph

    A student measures the current through a resistor at several potential differences. The points fall on a straight line through the origin, and the current is 0.30 A when the potential difference is 6.0 V. What is the resistance, and is the resistor ohmic?

    Show the solution
    1. Step 1: Slope of the I–ΔV graph = 0.30 A / 6.0 V = 0.050 A/V.
    2. Step 2: The slope is 1/R, so R = 1/0.050 = 20 Ω.
    3. Step 3: The graph is a straight line through the origin, meaning the resistance is constant, so the resistor is ohmic.

    Answer: 20 Ω; ohmic

Common mistakes

  • Reading the slope of an I–ΔV graph as R. The slope there is 1/R; it's R only when ΔV is on the vertical axis.
  • Using diameter instead of radius in A = πr², which makes the area four times too big.
  • Forgetting that stretching a wire also shrinks its cross-sectional area.
  • Confusing resistance (a property of one object) with resistivity (a property of the material).

On the exam

  • Lab questions often ask you to design an experiment to find resistivity, choose what to graph so it's linear, and use the slope.
  • Factor-of-change questions on R = ρℓ/A are common; track every quantity that changes.

Connected topics

Videos

Check yourself

4 questions on 11.3 Resistance, Resistivity, and Ohm’s Law. Pick an answer to see if you got it, and why.

Question 1 of 4Calculator allowed

A copper wire is 20 m long and has a diameter of 1.0 mm. Copper's resistivity is 1.7×10−81.7 \times 10^{-8} Ω·m. What is the wire's resistance?

Question 2 of 4Calculator allowed

A metal wire is stretched so that its length doubles while its volume stays the same. What happens to its resistance?

Question 3 of 4Calculator allowed

Wire 2 is made of the same metal as wire 1, but it is twice as long and has twice the diameter. How does the resistance of wire 2 compare with that of wire 1?

Question 4 of 4Calculator allowed

A student wants to find the resistivity of the graphite in a pencil lead of known, uniform diameter. She can connect the meters to different lengths of the lead. Which graph would let her find the resistivity most directly from its slope?

0 of 4 answered