A-level Physics (Advancing Physics)/Resistivity and Conductivity/Worked Solutions

'''1. A material has a conductivity of 106 S m−1. What is its resistivity?'''

$$\rho = \frac{1}{\sigma} = \frac{1}{10^6} = 10^{-6}\mbox{ }\Omega\mbox{ m} = 1\mbox{ }\mu\Omega\mbox{ m}$$

'''2. A pure copper wire has a radius of 0.5mm, a resistance of 1 MΩ, and is 4680 km long. What is the resistivity of copper?'''

$$\rho = \frac{10^6 \times \pi \times (0.5 \times 10^{-3})^2}{4680 \times 10^3} \approx 168 \times 10^{-9} \mbox{ }\Omega\mbox{ m} = 168\mbox{ n}\Omega\mbox{ m}$$

'''3. Gold has a conductivity of 45 MS m−1. What is the resistance of a 0.01m across gold connector, 0.05m long?'''

First, work out resistivity:

$$\rho = \frac{1}{45 \times 10^6} = 22.\bar{2} \times 10^{-9}\mbox{ }\Omega\mbox{m}$$

Then, substitute everything possible into the resistivity formula:

$$22.\bar{2} \times 10^{-9} = \frac{(\pi \times (0.5 \times 0.01)^2)R}{0.05} \approx 1.57 \times 10^{-3}R$$

$$R \approx \frac{22.\bar{2} \times 10^{-9}}{1.57 \times 10^{-3}} \approx 14.2 \times 10^{-6}\mbox{ }\Omega = 14.2\mbox{ }\mu\Omega\mbox{ }$$

'''4. A strand of metal is stretched to twice its original length. What is its new resistance? State your assumptions.'''

The material does not change, so resistivity is constant. Length doubles, and we know that volume must be constant.

V = AL

A = V/L

$$\rho = \frac{RA}{L} = \frac{R(\frac{V}{L})}{L} = \frac{RV}{L^2}$$

$$R_{old} = \frac{\rho}{1} \times \frac{L^2}{V} = \frac{\rho L^2}{V}$$

When we double L, we get:

$$R_{new} = \frac{\rho (2L)^2}{V} = \frac{4\rho L^2}{V} = 4 \times R_{old}$$

We are assuming that ρ and V are constant.

5. Which has the greater resistivity: a plank or a piece of sawdust, made from the same wood?

Sawdust and a plank are artefacts, not materials. Hence, they do not have a resistivity. Even if they did, they are made of the same thing, so they would have equal resistivity.