A dimensionless quantity is constructed in terms of electronic charge , permittivity of free space , Planck’s constant , and speed of light . If the dimensionless quantity is written as and is a non-zero integer, then is given by [2024]

Question

A dimensionless quantity is constructed in terms of electronic charge $e$ , permittivity of free space $ε_{0}$ , Planck’s constant $h$ , and speed of light $c$ . If the dimensionless quantity is written as $e^{α} ε_{0}^{β} h^{γ} c^{δ}$ and $n$ is a non-zero integer, then $(α, β, γ, δ)$ is given by [2024]

Smart Achievers · Accepted Answer

(1)

For dimensionless quantity

$e^{α} ε_{0}^{β} h^{γ} c^{δ} = M^{0} L^{0} T^{0} A^{0}$

Substituting dimensions of the physical quantities in the above relation and equating the dimensions of LHS and RHS

$\Rightarrow {(A T)}^{α} {(M^{- 1} L^{- 3} T^{4} A^{2})}^{β} {(M L^{2} T^{- 1})}^{γ} {(L T^{- 1})}^{δ} = A^{0} M^{0} L^{0} T^{0}$

$∴$ $α + 2 β = 0, α + 4 β - γ - δ = 0, - β + γ = 0, - 3 β + 2 γ + δ = 0$

$∴$ $α = - 2 β, β = γ, γ = δ$

$∴$ $α = 2 n, β = - n, γ = - n, δ = - n$

Solution

1 $(2 n, - n, - n, - n)$

2 $(n, - n, - 2 n, - n)$

3 $(n, - n, - n, - 2 n)$

4 $(2 n, - n, - 2 n, - 2 n)$

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Solution

Q. A dimensionless quantity is constructed in terms of electronic charge e, permittivity of free space ε0, Planck’s constant h, and speed of light c. If the dimensionless quantity is written as eαε0βhγcδ and n is a non-zero integer, then (α,β,γ,δ) is given by [2024]

1 (2n,-n,-n,-n)

2 (n,-n,-2n,-n)

3 (n,-n,-n,-2n)

4 (2n,-n,-2n,-2n)

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1 $(2 n, - n, - n, - n)$

2 $(n, - n, - 2 n, - n)$

3 $(n, - n, - n, - 2 n)$

4 $(2 n, - n, - 2 n, - 2 n)$