Question

The coefficient of superficial expansion of a solid is \[2 \times {10^{ - 5}}/^\circ C\]. Its coefficient of linear expansion is
A. \[4 \times {10^{ - 5}}/^\circ C\]
B. \[3 \times {10^{ - 5}}/^\circ C\]
C. \[2 \times {10^{ - 5}}/^\circ C\]
D. \[1 \times {10^{ - 5}}/^\circ C\]

Answer 1

Hint: Here, we will proceed by defining the terms coefficient of linear expansion and coefficient of superficial expansion. Then we will obtain the required answer by using the relation between coefficient of linear expansion and coefficient of superficial expansion.

Formula used:
\[\alpha = \dfrac{\beta }{2}\]

Complete step by step answer:
Coefficient of linear expansion: The ratio increases in length to original length for 1 degree rise in temperature is called the coefficient of linear expansion which is denoted by \[\alpha \].
Coefficient of superficial expansion: The ratio increases in area to original area for 1 degree rise in temperature is called the coefficient of superficial expansion which is denoted by \[\beta \].
We know that the coefficient of linear expansion is equal to half of the coefficient of superficial expansion. So, the relation is given by \[\alpha = \dfrac{\beta }{2}\].
Given the coefficient of superficial expansion of a solid \[\beta = 2 \times {10^{ - 5}}/^\circ C\]
Therefore, the coefficient of linear expansion is given by
\[
   \Rightarrow \alpha = \dfrac{{2 \times {{10}^{ - 5}}/^\circ C}}{2} \\
  \therefore \alpha = 1 \times {10^{ - 5}}/^\circ C \\
\]
Hence the coefficient of linear expansion of the solid is \[1 \times {10^{ - 5}}/^\circ C\]

Thus, the correct option is A. \[1 \times {10^{ - 5}}/^\circ C\].

Note:
Coefficient of superficial expansion is also called coefficient of areal expansion. There is another type of expansion which is called a coefficient of volume expansion which is defined by the ratio increase in volume to original volume for 1 degree rise in temperature and is denoted by \[\gamma \]. These three come under the types of thermal expansion and their relation is given by \[\alpha = \dfrac{\beta }{2} = \dfrac{\gamma }{3}\].