Question 1

The dimensions of the given quantities are: $[e] = [Q]$ $[\varepsilon_0] = [M^{-1}L^{-3}T^2Q^2]$ $[h] = [ML^2T^{-1}]$ $[c] = [LT^{-1}]$

The quantity is $e^\alpha \varepsilon_0^\beta h^\gamma c^\delta$. For it to be dimensionless: $[Q]^\alpha [M^{-1}L^{-3}T^2Q^2]^\beta [ML^2T^{-1}]^\gamma [LT^{-1}]^\delta = [M^0L^0T^0Q^0]$ Equating powers of dimensions: For $Q$: $\alpha + 2\beta = 0 \Rightarrow \alpha = -2\beta$ For $M$: $-\beta + \gamma = 0 \Rightarrow \gamma = \beta$ For $L$: $-3\beta + 2\gamma + \delta = 0 \Rightarrow -3\beta + 2\beta + \delta = 0 \Rightarrow \delta = \beta$ For $T$: $2\beta - \gamma - \delta = 0 \Rightarrow 2\beta - \beta - \beta = 0$ (This is consistent)

Thus, $\alpha = -2\beta$, $\gamma = \beta$, and $\delta = \beta$. Taking $\beta = -n$ (where $n$ is a non-zero integer), we get: $\alpha = 2n, \gamma = -n, \delta = -n$. So, $(\alpha, \beta, \gamma, \delta) = (2n, -n, -n, -n)$.