91Ó°ÊÓ

The coefficient of linear expansion (\(\alpha\)) is the change in length per unit length for a temperature change of 1°C. The coefficient of cubical expansion (\(\beta\)) gives the change in volume per unit volume for a temperature change of 1°C.

Given a substance with a linear, area, and volume increase of ΔL, ΔA, and ΔV, the increase in volume due to temperature change can be obtained by: \[ \beta = \frac{\Delta V}{V_0\Delta T} \] Where \(V_0\) is the initial volume, and \(\Delta T\) is the change in temperature.

When the length, width, and depth of a substance all increase with a linear expansion, the product of these increases is the cubical increase. Let's denote the increase in length, width, and depth as \(\Delta L_1, \Delta L_2,\) and \(\Delta L_3\), respectively. Then the cubical increase is given by: \[ \Delta V = \Delta L_1 \cdot \Delta L_2 \cdot \Delta L_3 \]

Using the coefficient of linear expansion \(\alpha\), we can express the increases in length, width, and depth as follows: \[ \Delta L_1 = \alpha L_1\Delta T \] \[ \Delta L_2 = \alpha L_2\Delta T \] \[ \Delta L_3 = \alpha L_3\Delta T \] Where \(L_1, L_2, L_3\) are the initial length, width, and depth, respectively.

Now substituting these linear increases into the cubical increase expression, we get: \[ \Delta V = (\alpha L_1\Delta T) \cdot (\alpha L_2\Delta T) \cdot (\alpha L_3\Delta T) \] \[ \Delta V = \alpha^3 L_1L_2L_3 (\Delta T)^3 \] Now, \(L_1L_2L_3 = V_0\), the initial volume of the substance, so we can rewrite the equation: \[ \Delta V = \alpha^3 V_0 (\Delta T)^3 \]

Using the equation for the coefficient of cubical expansion, we can rewrite the equation: \[ \beta = \frac{\Delta V}{V_0\Delta T} \] Substitute the expression for \(\Delta V\): \[ \beta = \frac{\alpha^3 V_0 (\Delta T)^3}{V_0\Delta T} \] This simplifies to: \[ \beta = \alpha^3 (\Delta T)^2 \]

Now we want to find the ratio of \(\beta\) to \(\alpha\). Dividing both sides of the equation by \(\alpha^3\), we get: \[ \frac{\beta}{\alpha^3} = (\Delta T)^2 \] Divide both sides by \((\Delta T)^2\): \[ \frac{\beta}{(\Delta T)^2}= \alpha^3 \] Now take the cube root of both sides of the equation: \[ \sqrt[3]{\frac{\beta}{(\Delta T)^2}}= \alpha \] Now to find the ratio of \(\alpha : \beta\): \[ \frac{\alpha}{\beta} = \frac{\sqrt[3]{\frac{\beta}{(\Delta T)^2}}}{\beta} \] \[ \frac{\alpha}{\beta} = \frac{1}{\beta^{2/3} (\Delta T)^{4/3}} \] The ratio of coefficients of cubical expansion and linear expansion is not constant and depends on the change in temperature and the cubical coefficient of expansion. Therefore, the correct answer is: (D) None of these