JEE Advanced · 2024 · 26 May 2024 · Paper 1 · Units & Measurement

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

Answer: (A)

Solution

For the quantity to be dimensionless, we require

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

. Substituting dimensions:

(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}

. Equating powers:

α + 2 β = 0

α + 4 β - γ - δ = 0

- β + γ = 0

- 3 β + 2 γ + δ = 0

. This gives

α = - 2 β

β = γ

γ = δ

. Option (A) with

α = 2 n

β = - n

γ = - n

δ = - n

satisfies all conditions.

JEE Advanced · 2024 · 26 May 2024 · Paper 1 · Single correct option