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A dimensionless quantity is constructed in terms of electronic charge $e$, permittivity of free space $\varepsilon_0$, Planck's constant $h$, and speed of light $c$. If the dimensionless quantity is written as $e^\alpha \varepsilon_0^\beta h^7 c^5$ and $n$ is a non-zero integer, then $(\alpha, \beta, \gamma, \delta)$ is given by
$(2 n,-n,-n,-n)$
$(n,-n,-2 n,-n)$
$(n,-n,-n,-2 n)$
$(2 n,-n,-2 n,-2 n)$
Solution
For the quantity to be dimensionless
$e ^\alpha \varepsilon_0^\beta h ^\gamma c ^{ d }= M ^0 L ^0 T ^0 A ^0$
$\Rightarrow( AT )^\alpha\left( M ^{-1} L ^{-3} T ^4 A ^2\right)^\beta\left( ML ^2 T ^{-1}\right)^\gamma\left( LT ^{-1}\right)^\delta= A ^0 M ^0 L ^0 T ^0$
$\therefore \alpha+2 \beta=0, \alpha+4 \beta-\gamma-\delta=0,-\beta+\gamma=0 \&-3 \beta+2 \gamma+\delta=0$
$\therefore \alpha=-2 \beta, \beta=\gamma \& \gamma=\delta$
$\therefore$ Option $(A)$ satisfies the given condition
