The increase or expansion of the length of a solid due to heating is called linear expansion.
Let us consider a solid rod of length 5cm(l1) at temperature 10.C(θ1). Suppose the rod expands to 10cm(l2) when it is heated up to 20.C(θ2) . Then the change in length(∆l) is observed to be 5cm. It has been experimentally observed that this change in length is:
1. Directly proportional to the original length. That is, if the original length of the solid is more then the change in length of the solid will also be more.
2. Directly proportional to the change in temperature. That is, if the change in temperature that happens due to heating is more then the change is length is also more.
Now, combining equation (1) and (2), we get,
∆l ∝ l1(θ2-θ1)
or, ∆l = α l1(θ2-θ1).........(3)
where α is a proportionality constant known as coefficient of linear expansion or linear expansivity.
Linear expansivity is literally the coefficient of linear expansion. Its value depends upon the nature of material. It can be also said that α is same for all solids made up of same material.
Using equation (3)
∆l = l1(θ2-θ1)
or, α = ∆l / l1(θ2-θ1)
For l1=1 m and (θ2-θ1)=1.C or 1.K ,
Thus, linear expansivity of the material of a rod is defined as the change in length per unit original length per unit change in temperature.
The SI unit of α is .C-1 or .K-1.
Now, we know. ∆l = αl1(θ2-θ1)
or, l2-l1= αl1(θ2-θ1) [since, ∆l is change in length which is l2-l1]
or, l2= l1 + αl1(θ2-θ1)
or, l2 = l1 [1 + α(θ2-θ1)] (taking common)
This equation is the required equation to find the length of solid after expansion.