To solve this problem, we need to understand the relationship between the coefficient of volume expansion and the coefficient of areal expansion for a solid material.
1. Coefficient of Volume Expansion (α_v): This is a measure of the fractional change in volume per degree change in temperature. Given in the problem as:
[tex]\[
\alpha_v = 0.00027 /^{\circ} C
\][/tex]
2. Coefficient of Areal Expansion (α_a): This is the measure of the fractional change in area per degree change in temperature. For isotropic materials (materials with properties that are the same in all directions), the coefficient of areal expansion can be approximated as two-thirds of the coefficient of volume expansion. Mathematically, this relationship is given by:
[tex]\[
\alpha_a = \frac{2}{3} \alpha_v
\][/tex]
3. Calculation:
[tex]\[
\alpha_a = \frac{2}{3} \times 0.00027 /^{\circ} C
\][/tex]
When we multiply 0.00027 by [tex]\(\frac{2}{3}\)[/tex], we get:
[tex]\[
\alpha_a = 0.00018 /^{\circ} C
\][/tex]
Evaluating the options given:
1. [tex]$0.00009 /^{\circ} C$[/tex]
2. [tex]$0.00018 /^{\circ} C$[/tex]
3. [tex]$0.00027 /^{\circ} C$[/tex]
4. [tex]$0.00003 /^{\circ} C$[/tex]
The correct coefficient of areal expansion is [tex]\(0.00018 /^{\circ} C\)[/tex].
Thus, the correct answer is:
2) [tex]$0.00018 /^{\circ} C$[/tex]
