Question

In an anisotropic medium, the coefficients of linear expansion of a solid are ${{\alpha }_{1}},{{\alpha }_{2}}$ and ${{\alpha }_{3}}$ in three mutually perpendicular directions. The coefficient of volume expansion for the solid is
A. ${{\alpha }_{1}}-{{\alpha }_{2}}+{{\alpha }_{3}}$
B. $\dfrac{{{\alpha }_{1}}+{{\alpha }_{2}}+{{\alpha }_{3}}}{3}$
C. ${{\alpha }_{1}}+{{\alpha }_{2}}+{{\alpha }_{3}}$
D. None of these

Answer 1

As a very first step, you could consider a cuboid and then fix the temperature to rise from $0{}^\circ$ to $T{}^\circ$. Then, you could find the expression for linear expansion along the three directions. Then by taking their product we will arrive at the volume of the solid. Now by expanding that expression and then neglecting the smaller terms you will arrive at an expression. Comparing that with the expression for volume expansion you will get the answer.

Formula used:
Linear expansion,
$l={{l}_{0}}\left( 1+\alpha \Delta T \right)$
Volume expansion,
$V={{V}_{0}}\left( 1+\gamma \Delta T \right)$

Complete step by step solution:
Let us consider a solid of cuboidal shape.

Let ${{\alpha }_{1}},{{\alpha }_{2}},{{\alpha }_{3}}$ be the coefficients of linear expansions along edges ${{a}_{1}},{{a}_{2}},{{a}_{3}}$ respectively.
For linear expansion, we have,
$l={{l}_{0}}\left( 1+\alpha \Delta T \right)$
Let the temperature of the above solid increase from $0{}^\circ C$ to$T{}^\circ C$. So,
$\Delta T=T-0=T$
Applying this formula for all the three edges we have,
${{a}_{1}}'={{a}_{1}}\left( 1+{{\alpha }_{1}}T \right)$……………………………………. (1)
${{a}_{2}}'={{a}_{2}}\left( 1+{{\alpha }_{2}}T \right)$……………………………………. (2)
${{a}_{3}}'={{a}_{3}}\left( 1+{{\alpha }_{3}}T \right)$……………………………………. (3)
The product of equations (1), (2)and (3) will give the volume of the cuboid at temperature T after expansion. That is,
$V'={{a}_{1}}'{{a}_{2}}'{{a}_{3}}'={{a}_{1}}{{a}_{2}}{{a}_{3}}\left( 1+{{\alpha }_{1}}T \right)\left( 1+{{\alpha }_{2}}T \right)\left( 1+{{\alpha }_{3}}T \right)$ ……………………………… (4)
But we know that the product of the edges will give the initial volume of the cuboid. That is,
${{V}_{0}}={{a}_{1}}{{a}_{2}}{{a}_{3}}$
Now, after expansion of equation (4) and then by neglecting the negligibly small values, we get,
$\Rightarrow V'\approx {{V}_{0}}\left( 1+\left( {{\alpha }_{1}}+{{\alpha }_{2}}+{{\alpha }_{3}} \right)T \right)$ ……………………………………. (5)
Now, we have the following formula for volume expansion due to increase in temperature,
$V={{V}_{0}}\left( 1+\gamma \Delta T \right)={{V}_{0}}\left( 1+\gamma T \right)$ ……………………………………………. (6)
Comparing equation (5) with equation (6) we get the coefficient of volume expansion $\gamma$ as,
$\gamma ={{\alpha }_{1}}+{{\alpha }_{2}}+{{\alpha }_{3}}$
Hence, option C is the correct answer.

Note: Thermal expansion is the term given to the tendency of matter to undergo change in length, area and volume as a result of being subjected to change in temperature. On being heated the molecules move and vibrate more thereby increasing the distance between them. The coefficient of thermal expansion can be defined as the ratio of relative expansion to the temperature change.