JEE Advanced 2024 — Paper 1 — PHY-I solution

JEE Advanced2024Paper 1PHY-I

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

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

Solution

Write each quantity in terms of base SI dimensions:

$[e] = [AT]$ , $[ε_{0}] = [M^{- 1} L^{- 3} T^{4} A^{2}]$ , $[h] = [ML^{2} T^{- 1}]$ , $[c] = [LT^{- 1}]$ .

For the combination $e^{α} ε_{0}^{β} h^{γ} c^{δ}$ to be dimensionless, the exponent of every base dimension must vanish:

Mass: $- β + γ = 0$

Length: $- 3 β + 2 γ + δ = 0$

Time: $α + 4 β - γ - δ = 0$

Current: $α + 2 β = 0$

From the current equation, $α = - 2 β$ . From the mass equation, $γ = β$ . Substituting into the length equation gives $δ = β$ .

Setting $β = - n$ (so that $n$ is a non-zero integer parameter) yields $(α, β, γ, δ) = (2 n, - n, - n, - n)$ .

$∴$ Option (A) is correct.

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