Question

An isotropic solid has linear expansion (coefficient of \[{{\alpha }_{x}}\],\[{{\alpha }_{y}}\]and \[{{\alpha }_{z}}\]for three rectangular axes in a solid). The coefficient of cubical expansion is
\[\begin{align}
  & A.\,{{\alpha }_{x}}{{\alpha }_{y}}{{\alpha }_{z}} \\
 & B.\,\dfrac{{{\alpha }_{x}}}{{{\alpha }_{y}}+{{\alpha }_{z}}} \\
 & C.\,{{\alpha }_{x}}+{{\alpha }_{y}}+{{\alpha }_{z}} \\
 & D.\,{{\alpha }^{2}}_{x}+{{\alpha }^{2}}_{y}+{{\alpha }^{2}}_{z} \\
\end{align}\]

Answer 1

Hint: The coefficient of the cubical expansion equals the coefficient of the volume expansion. As the coefficients of the rectangular solid are given, so, we will consider the cuboid to find the coefficients of the cubical expansion.
Formula used:
\[V={{V}_{0}}(1+\gamma T)\]

Complete answer:
From the given information, we have the data as follows.
An isotropic solid has linear expansion (coefficient of \[{{\alpha }_{x}}\],\[{{\alpha }_{y}}\]and \[{{\alpha }_{z}}\]for three rectangular axes in a solid).
The initial temperature of the cuboid is, \[0{}^\circ C\]
The final temperature of the cuboid is, \[T{}^\circ C\]
Thus, the change in the temperature is,
 \[\begin{align}
  & \Delta T=T-0 \\
 & \therefore \Delta T=T{}^\circ C \\
\end{align}\]
The new linear expansion coefficients at the temperature T are\[\alpha {{'}_{x}}\],\[\alpha {{'}_{y}}\]and \[\alpha {{'}_{z}}\].
\[\begin{align}
  & a_{x}^{'}={{a}_{x}}(1+{{\alpha }_{x}}T) \\
 & a_{y}^{'}={{a}_{y}}(1+{{\alpha }_{y}}T) \\
 & a_{z}^{'}={{a}_{z}}(1+{{\alpha }_{z}}T) \\
\end{align}\]
The coefficient of the cubical expansion equals the coefficient of the volume expansion.
The volume of the cuboid at the temperature T is \[a_{x}^{'}a_{y}^{'}a_{z}^{'}\]
Consider the volume of the cuboid at this temperature and substitute the values of the linear expansion coefficients.
As the coefficients of the rectangular solid are given, so, we will consider the cuboid to find the coefficients of the cubical expansion.
\[\begin{align}
  & V'=a_{x}^{'}a_{y}^{'}a_{z}^{'} \\
 & \Rightarrow V'={{a}_{x}}(1+{{\alpha }_{x}}T)\times {{a}_{y}}(1+{{\alpha }_{y}}T)\times {{a}_{z}}(1+{{\alpha }_{z}}T) \\
 & \therefore V'={{a}_{x}}{{a}_{y}}{{a}_{z}}(1+{{\alpha }_{x}}T){{a}_{y}}(1+{{\alpha }_{y}}T){{a}_{z}}(1+{{\alpha }_{z}}T) \\
\end{align}\]
The coefficient of the volume expansion is, \[V={{V}_{0}}(1+\gamma T)\]
Continue further expansion of the above equation.
\[V'={{a}_{x}}{{a}_{y}}{{a}_{z}}\left[ 1+({{\alpha }_{x}}+{{\alpha }_{y}}+{{\alpha }_{z}})T+{{\alpha }_{x}}{{\alpha }_{y}}{{\alpha }_{z}}{{T}^{3}}+{{T}^{2}}({{\alpha }_{x}}{{\alpha }_{y}}+{{\alpha }_{y}}{{\alpha }_{z}}+{{\alpha }_{x}}{{\alpha }_{z}}) \right]\]
Neglect the \[{{T}^{2}}\]and \[{{T}^{3}}\]terms as, \[{{\alpha }_{x}}^{2}<<<{{\alpha }_{x}}\Rightarrow {{\alpha }_{x}}{{\alpha }_{y}}<<<{{\alpha }_{x}},\,\,{{\alpha }_{x}}{{\alpha }_{y}}{{\alpha }_{z}}<<<{{\alpha }_{x}}\]
\[V'={{a}_{x}}{{a}_{y}}{{a}_{z}}\left[ 1+({{\alpha }_{x}}+{{\alpha }_{y}}+{{\alpha }_{z}})T \right]\]
Compare the above equation with the basic equation of the volume expansion. So, we get,
\[\gamma ={{\alpha }_{x}}+{{\alpha }_{y}}+{{\alpha }_{z}}\]
\[\therefore \] The coefficient of cubical expansion is \[{{\alpha }_{x}}+{{\alpha }_{y}}+{{\alpha }_{z}}\] .

Thus, option (C) is correct.

Note:
The basic relation between the coefficient of the linear expansion and the coefficient of the volume expansion is that the coefficient of the volume expansion is 3 times the coefficient of the linear expansion. Both these coefficients depend on the temperature and the type of material.