Define co-efficient of linear and volume expansions in thermal expansion and establish the relation between them.

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> [!TLDR]
> $\alpha = \Delta L/(L\Delta T)$, $\gamma = \Delta V/(V\Delta T)$, and for an isotropic solid $\gamma = 3\alpha$.
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> **Coefficient of linear expansion (α):** When a solid rod of original length L is heated through a temperature rise $\Delta T$, its length increases by $\Delta L$. The coefficient of linear expansion is defined as the fractional increase in length per unit rise in temperature:
> $$\alpha = \frac{\Delta L}{L\,\Delta T}$$
> SI unit: K⁻¹.
>
> **Coefficient of volume (cubical) expansion (γ):** Similarly, if a solid of original volume V expands by $\Delta V$ when heated through $\Delta T$, the coefficient of volume expansion is:
> $$\gamma = \frac{\Delta V}{V\,\Delta T}$$
> SI unit: K⁻¹.
>
> **Deriving the relation γ = 3α:** Consider a cube of an isotropic solid with initial side length L. On heating through $\Delta T$, each side expands equally (isotropic ⇒ same α in every direction) to a new length:
> $$L' = L(1+\alpha\Delta T)$$
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> The new volume is:
> $$V' = L'^3 = L^3(1+\alpha\Delta T)^3$$
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> Expanding using the binomial theorem and neglecting the very small higher-order terms ($\alpha^2, \alpha^3$, since $\alpha\Delta T \ll 1$): …

Q.Define co-efficient of linear and volume expansions in thermal expansion and establish the relation between them.

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Q.Define co-efficient of linear and volume expansions in thermal expansion and establish the relation between them.

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