If it is known that a dimensionless number significant in heat energy diffusion in case of convective heat transfer is given by $\frac{A^{a} B}{C}$ then the value of a is, (dimension of $A$ = $M L^{2} T^{- 2} K^{- 1}$ , $B$ = $L^{- 1} T^{- 1}$ , $C$ = $M L^{1} T^{- 3} K^{- 1}$ )

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Solution

The correct option is D 1
Let, $x = \frac{A^{a} B}{C}$
As $x$ is a dimensionless quantity so we can write ,
$[M^{0} L^{0} T^{0} K^{0}] = \frac{[M^{a} L^{2 a} T^{- 2 a} K^{- a}] [L^{- 1} T^{- 1}]}{[M L^{1} T^{- 3} K^{- 1}]} = [M^{(a - 1)} L^{(2 a - 1 - 1)} T^{(- 2 a - 1 + 3)} K^{(- a + 1)}]$

Comparing the power from both sides, $0 = a - 1$ or $a = 1$

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