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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.


       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∝(θ21) ...........(2)

Now, combining equation (1) and (2), we get,

     ∆l  ∝  l121)

or, ∆l = α l121).........(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(θ21

or, α = ∆l / l121)

For l1=1 m and (θ21)=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 = αl121)

or, l2-l1= αl121) [since, ∆l is change in                                        length which is l2-l1]

or, l2= l1 + αl121)

or, l2 = l1 [1 + α(θ21)] (taking common)

This equation is the required equation to find the length of solid after expansion.