This is an old revision of the document!
Resistance of a Wire
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 50 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$^2$?
Facts
- The original wire has $L = 2 \text{ cm}$, $A = 1 \text{ mm}^2$, and $R = 50 \text{ m}Omega$.
- The length could be increased to $L_{new} = 4 \text{ cm}$.
- The cross-sectional area could be increased to $A_{new} = 3 \text{ mm}^2$.
Lacking
- Resistances of new wires.
Approximations & Assumptions
- The conductivity of the wire does not change.
- The wire's material is uniform.
Representations
- We represent the resistance of a simple wire such as this with: $$R = \frac{L}{\sigma A}$$
Solution
All we need here is our representation for the resistance of the wire. In the first change to the wire, we trible it's length ($2 \text{ cm} \rightarrow 6 \text{ cm}$). Our new resistance then is found by $$R_{new} = \frac{L_{new}}{\sigma A} = \frac{3L}{\sigma A} = 3R = 150 \text{ m}\Omega$$