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^gamma c^delta ) and ( n ) is a non-zero integer, then ( (alpha, beta, gamma, delta) ) is given by[latexpage]

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Problem:

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 $e^\alpha \varepsilon_0^\beta h^\gamma c^\delta$ and $n$ is a non-zero integer, then $(\alpha, \beta, \gamma, \delta)$ is given by:

(A) $(2n, -n, -n, -n)$ (B) $(n, -n, -2n, -n)$

Context:

A dimensionless quantity is expressed using the fundamental physical constants:

Electronic charge $e$

Permittivity of free space $\varepsilon_0$

Planck’s constant $h$

Speed of light $c$

The given expression is in the form:

$e^{\alpha} \varepsilon_0^{\beta} h^{\gamma} c^{\delta}$

This is said to be dimensionless, meaning its overall dimensional formula must be $[M^0L^0T^0]$ . The task is to determine the correct combination $(\alpha, \beta, \gamma, \delta)$ in terms of a non-zero integer $n$ , based on the options provided.

Explanation:

To find the powers $\alpha, \beta, \gamma, \delta$ , the dimensions of each constant are substituted and the resulting expression is equated to zero in each base dimension (Mass – $M$ , Length – $L$ , Time – $T$ , Ampere – $A$ ) to solve the system of equations: