Linear Coefficient of expansion

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.

           ∆l∝l₁..........(1)

       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.

              ∆l∝(θ₂-θ₁) ...........(2)

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(α):

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 ,

α=∆l

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.