GATE 2023 PH – Question 12

Correct answer: (D) Fine structure constant

Explanation

A **dimensionless constant** is a pure number, independent of the system of units.

• **Permittivity of free space** $\epsilon_0$: units C² N⁻¹ m⁻². ✗
• **Permeability of free space** $\mu_0$: units N A⁻². ✗
• **Bohr magneton** $\mu_B=e\hbar/2m_e$: units J/T (A m²). ✗
• **Fine structure constant** $\alpha=\dfrac{e^2}{4\pi\epsilon_0\hbar c}\approx\dfrac1{137}$: the units of $e^2/(\epsilon_0)$ (J m) cancel with those of $\hbar c$ (J m), so it is a pure number. ✓

Answer **fine structure constant** (D).