When you heat a solid object, its atoms and molecules vibrate more vigorously. This increased motion causes them to push their neighbours further apart, leading to an overall expansion of the material. Let's first consider this expansion along a single dimension, like the length of a metal rod. The change in length, denoted as $ΔL$, is directly proportional to the original length $L₀$ and the change in temperature $ΔT$.

To turn this proportionality into an equation, we introduce a constant specific to the material, called the coefficient of linear expansion, symbolised by $α$ (alpha). This coefficient tells us how much a material expands per unit length for each degree of temperature change.

The unit for $α$ is per degree Celsius (/$°$C) or per Kelvin (/K). A material with a high $α$, like aluminium, expands more than a material with a low $α$, like steel, for the same temperature increase. This principle is why engineers leave small gaps in railway tracks and concrete slabs. Without these expansion gaps, the immense forces generated by thermal expansion on a hot day could cause the tracks to buckle or the concrete to crack.

Expansion doesn't just happen in a straight line. When a two-dimensional plate or a three-dimensional block is heated, it expands in all directions. We can describe these changes using similar coefficients.

For a two-dimensional object like a metal sheet, we use the coefficient of superficial (or area) expansion, $β$ (beta). The change in area ($ΔA$) is proportional to the original area $A₀$ and the temperature change $ΔT$.

For a three-dimensional object, we use the coefficient of volume expansion, $γ$ (gamma). This describes the change in the object's total volume. The formula is analogous.

You might have noticed that $α$, $β$, and $γ$ are not independent. They are directly related. For most solids, which expand uniformly in all directions (isotropically), we can derive a simple relationship between them.

Consider a square sheet with an initial side length of $L₀$. Its initial area is $A₀ = L₀²$. After heating by $ΔT$, the new side length is $L = L₀(1 + αΔT)$. The new area will be: $A = L² = [L₀(1 + αΔT)]² = L₀²(1 + 2αΔT + α²(ΔT)²)$

Since the value of $α$ is very small (typically around $10⁻⁵$ /°C), the $α²$ term is extremely small and can be safely ignored. So, we get: $A ≈ L₀²(1 + 2αΔT) = A₀(1 + 2αΔT)$

Comparing this with our area expansion formula, $A = A₀(1 + βΔT)$, we can see that $β ≈ 2α$.

We can apply the same logic to a cube with an initial side length $L₀$ and volume $V₀ = L₀³$. The new volume is: $V = L³ = [L₀(1 + αΔT)]³$

Using the binomial expansion and ignoring the very small terms with $α²$ and $α³$, we get: $V ≈ L₀³(1 + 3αΔT) = V₀(1 + 3αΔT)$

Comparing this with the volume expansion formula, $V = V₀(1 + γΔT)$, we find that $γ ≈ 3α$. This gives us a very useful relationship for isotropic materials.

This relationship is what allows a bimetallic strip to work. It is made of two different metals, like steel and brass, bonded together. Since brass has a higher coefficient of expansion than steel, it expands more when heated, causing the strip to bend.

Let's apply these concepts to a typical problem. Suppose a steel ruler is exactly 50 cm long at a temperature of 20 °C. What will its length be on a hot day when the temperature is 45 °C? The coefficient of linear expansion for steel is $1.2 \times 10⁻⁵$ /°C.

The ruler is now 50.015 cm long. While the change seems small, over large structures and with large temperature swings, these effects become critically important for engineers to consider.