Hello students, we know that when heat is provided to substance, most substances gets expanded on heating and contracted on cooling. Due to change in temperature strain is produced in substance which c causes change in its dimensions. The increase in the dimensions of a body due to the increase in its temperature is called thermal expansion.

When heat is given to substance three important types of expansion are produced, here we are going to discuss the expansion related to change in area of substance, i.e. areal expansion.

**Let’s understand concept of areal expansion in details….!**

**Increase in area of substance under action of heat in known as areal expansion.**

Consider a thin uniform metal rod of initial area A_{0} at 0^{0} C. Let that rod is being heated to t^{0} C, so that its final area becomes A_{t}. As we know that, here the expansion is a long the area of rod, then the increase in area of rod is given as,

∴ increase in area = A_{t} – A_{0} ……………………….(1)

We can easily observed that when this increase in area depends upon following factors,

- Increase in area is directly proportional to area at 0
^{0}C i.e. A_{0} - Increase in area is directly proportional to rise in temperature i.e. ∆t

Then mathematically we can write as,

∴ A_{t} – A_{0} α A_{0}

∴ A_{t} – A_{0} α ∆t

∴ A_{t }– A_{0} =β A_{0}.∆t…………… (2)

Where β is constant of proportionality, here known as coefficient of areal expansion.

Then from above equation we can write as,

Hence ‘the coefficient of areal expansion is defined as the increase in area of substance per unit area at 0^{0} C per unit rise in temperature ∆t’

SI unit of coefficient of areal expansion is per ^{0}C or /^{0}C

From equation (2) we can get, expression for final area of rod as,

∴A_{t }– A_{0} = β A_{0}.∆t

∴A_{t} = A_{0}+β L_{0}.∆t

∴A_{t} = A_{0} (1+β .∆t) ……………………….(4)

The above formula is for the final area of material of rod when its temperature is increased by t^0 C.

If β_{1} and β_{2} be the initial and final areas of rod when heated from at temperature t_{1}^{0} C to t_{2}^{0} C, then we can write,

∴A_{2} = A_{1} [1+β.(t_{2} – t_{1})]

**Lets solve some numericals…..!**

**Ex:1) Increase in Area of metal rod is 0.25 mm ^{2} if its initial area is 55 cm^{2}. Find the coefficient of areal expansion when heated from 35^{0} C to 180^{0} C.**

Solution:

Here, A_{2} – A_{1} = 0.25 mm^{2} = 25 × 10^{-9} m^{2}, t_{1}= 35^{0} C ,t_{2}= 180^{0} C,

A_{1}= 55 × 10^{-4} m^{2}

∴∆t = t_{2} – t_{1}= 145^{0} C