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

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

TestHub · Accepted Answer

For the quantity to be dimensionless ​ e α ε 0 β ​ h γ c d = M 0 L 0 T 0 A 0 ⇒ ( AT ) α ( M − 1 L − 3 T 4 A 2 ) β ( ML 2 T − 1 ) γ ( LT − 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 ​ ∴ Option (1) satisfies the given condition

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