Question

Ratio among linear expansion coefficient ($\alpha$), aerial expansion coefficient ($\beta$) and volume expansion coefficient ($\gamma$) is :
$(a).\,1:2:3\\\ (b).\,3:2:1\\\ (c).\,4:3:2\\\ (d).\,{\rm{None of these}}$

Answer 1

In this question, we have to use the relation between among linear expansion coefficient ($\alpha$), aerial expansion coefficient ($\beta$) and volume expansion coefficient($\gamma$),
$\alpha = \dfrac{\beta }{2} = \dfrac{\gamma }{3}$.
With the above relation we can calculate the relation between them individually.

Complete step by step answer:
For this we will take one of them to be constant, for if we take $\alpha$.
Taking $\alpha$ constant,
$\alpha \, = \,\dfrac{\beta }{2}$, therefore, after multiplying 2 to both sides $\beta = 2\alpha$
Similarly, we calculate $\gamma$,
$\alpha \,\,\,\, = \,\dfrac{\gamma }{3}\\\ \,3\alpha \, = \gamma$
Now we will put the above calculated values in the ratio of the three coefficients of expansions, and cancel the variable $\alpha$

$$\alpha :\beta :\gamma = \alpha :2\alpha :3\alpha \\\ \alpha :\beta :\gamma = 1:2:3$$

Therefore, the ratio is 1:2:3 and the correct option is (a).
Additional Information:
Thermal expansion occurs in matter due to change in volume when subjected to change in temperature. The change in length of a body due to thermal expansion is related to temperature by a linear expansion coefficient. The areal expansion coefficient is proportional to the change in body’s area dimensions with a change in temperature (It is the fractional change in area per degree of temperature change). The most basic expansion due to heat is volumetric thermal expansion. The body expands and contracts in all the dimensions when subjected to a temperature change. All three coefficients of thermal expansions are related to each other.

Note: The relation among linear expansion coefficient, aerial expansion coefficient and volume expansion coefficient is to be remembered for the questions like this. Taking one of the values as constant and converting the other two in that variable should be done for taking out the ratio