Animated Solution for Physics - Current Electricity: The resistance of a conductor at $15^{\circ }\text{C}$ is $16\Omega$ and at $100^{\circ }\text{C}$ is $20\Omega$. What will be the temperature coefficient of resistance of the conductor?

Imagine you are heating up a piece of metal wire. As the temperature rises, the atoms inside the metal start vibrating more vigorously. These vibrating atoms scatter the flowing electrons more frequently, which means the resistance of the wire increases! For most standard conductors over a reasonable temperature range, this increase is beautifully linear.

The master equation that governs this physical reality is:

Here, $R_t$ is the resistance at some temperature $t$, $R_0$ is the resistance at a reference temperature (usually $0^{\circ }\text{C}$), $\alpha$ is the temperature coefficient of resistance, and $\Delta T$ is the change in temperature from the reference.

In our problem, we are given two distinct states of the conductor. First, at $15^{\circ }\text{C}$, the resistance is $16\Omega$. Let's plug this into our master equation, assuming our reference temperature is $0^{\circ }\text{C}$:

Next, we are told that at $100^{\circ }\text{C}$, the resistance climbs to $20\Omega$. Setting up the second equation, we get:

We now have a system of two equations with two unknowns: $R_0$ and $\alpha$.

Since we are only interested in finding $\alpha$, the most elegant way forward is to eliminate $R_0$. How do we do that? By simply dividing the first equation by the second!

The $R_0$ terms cancel out perfectly. Simplifying the fraction on the left gives us:

Now, it's just a matter of careful algebra. Let's cross-multiply to get rid of the fractions:

Expanding the brackets, we get:

Let's group all the $\alpha$ terms on one side and the constants on the other:

Finally, isolating $\alpha$, we find:

When you perform this division, you get:

And there you have it! The temperature coefficient of resistance for this conductor is $0.003^{\circ }{\text{C}}^{-1}$. This tells us exactly how much the resistance will fractionally increase for every single degree rise in temperature.