When a metal rod is heated, its resistance changes both because of a change in resistivity and because of a change in the length of the rod. If a silver rod has a resistance of 1.70 Ω at 21.0°C, what is its resistance when it is heated to 180.0°C? The temperature coefficient for silver is α = 6.1 ✕ 10−3 °C−1, and its coefficient of linear expansion is 18 ✕ 10−6 °C−1. Assume that the rod expands in all three dimensions.

Explanation:

The resistance of a metal rod is given by

where

is the resistivity

L is the length of the rod

A is the cross-sectional area

The resistivity changes with the temperature as:

where in this case:

is the resistivity of silver at

is the temperature coefficient for silver

is the current temperature

Substituting,

The length of the rod changes as

where:

is the initial length at

is the coefficient of linear expansion

Substituting,

The cross-sectional area of the rod changes as

So, substituting,

Therefore, if the initial resistance at 21.0°C is

Then the resistance at 180.0°C is:

resonant frequency