Suppose you have a wire whose resistance you know. The wire has a length of 2 cm, and has a cross-sectional area of 1 mm$^2$. The resistance of the wire is 60 m$\Omega$. What is the resistance if you increase the length of the wire to 6 cm? What if you increase the cross-sectional area to 3 mm$^2$?

• The original wire has $L = 2 \text{ cm}$, $A = 1 \text{ mm}^2$, and $R = 60 \text{ m}\Omega$.
• The length could be increased to $L_{new} = 6 \text{ cm}$.
• The cross-sectional area could be increased to $A_{new} = 3 \text{ mm}^2$.

• Resistances of new wires.

• The conductivity of the wire does not change.
• The wire's material is uniform.

• We represent the resistance of a simple wire such as this with: $$R = \frac{L}{\sigma A}$$

All we need here is our representation for the resistance of the wire. In the first change to the wire, we triple it's length ($2 \text{ cm} \rightarrow 6 \text{ cm}$). Our new resistance then is found by $$R_{new} = \frac{L_{new}}{\sigma A} = \frac{3L}{\sigma A} = 3R = 180 \text{ m}\Omega$$

If instead, we made the other change, we would have tripled the cross sectional area ($1 \text{ mm}^2 \rightarrow 3 \text{ mm}^2$). Our new resistance would then be $$R_{new} = \frac{L}{\sigma A_{new}} = \frac{L}{\sigma 3 A} = \frac{1}{3}R = 20 \text{ m}\Omega$$