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**Hint:**To understand this question, we have to consider a sample body and then study the expansion in it by application of heat. The expansion in the body happens in three ways: linear, superficial and volumetric. Their definitions must be understood to solve this problem.

**Complete step by step answer:**

There are three kinds of expansion of a solid when heat is applied to it:

i) Linear: Change in the length

ii) Superficial or Areal: Change in the area

iii) Volumetric: Change in the volume.

To understand the expansion, let us take an example of a solid rectangle of dimensions a, b, and c heated from ${0^ \circ }C$ to a temperature ${t^ \circ }C$ where the new dimensions are A, B and C.

The original volume of the solid at ${0^ \circ }C$ = $abc$

The volume of the solid at temperature ${t^ \circ }C$ = ABC

The coefficient of linear expansion is represented by $\alpha $. If A is the length at temperature ${t^ \circ }C$ and a is the length at ${0^ \circ }C$, the coefficient of linear expansion is given by the relation –

$A = a\left( {1 + \alpha t} \right)$

Similarly, for the other dimensions we have –

$B = b\left( {1 + \alpha t} \right)$

$C = c\left( {1 + \alpha t} \right)$

Final volume at temperature ${t^ \circ }C$ –

\[ABC = a\left( {1 + \alpha t} \right) \times b\left( {1 + \alpha t} \right) \times c\left( {1 + \alpha t} \right)\]

\[ \Rightarrow ABC = abc{\left( {1 + \alpha t} \right)^3}\]

\[ \Rightarrow ABC = abc\left( {1 + 3\alpha t + 3{\alpha ^2}{t^2} + {\alpha ^3}{t^3}} \right)\]

The value of coefficient of linear expansion is very low. Hence, the higher powers of $\alpha $ are negligible.

Therefore, we have –

\[ABC = abc\left( {1 + 3\alpha t} \right)\]

The coefficient of volumetric expansion is represented by $\gamma $. If V is the length at temperature ${t^ \circ }C$ and v is the length at ${0^ \circ }C$, the coefficient of linear expansion is given by the relation –

$V = v\left( {1 + \gamma t} \right)$

Comparing this equation with the above, we get the following relationship between the coefficients as –

$\gamma = 3\alpha $

$\therefore \alpha = \dfrac{\gamma }{3}$

Hence, the coefficient of linear expansion is one-third the coefficient of volumetric expansion.

**Hence, the correct option is Option B.**

**Note:**There is another coefficient of expansion known as the coefficient of superficial expansion represented by $\beta $. The one equation relating all the three coefficients of expansion is:

$\dfrac{\alpha }{1} = \dfrac{\beta }{2} = \dfrac{\gamma }{3}$

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