JEE PYQ: Units & Measurements - Question ID a76d50f6650f (JEE Main 2021)

In this question we match physical constants (List-I) with their dimensional formulae (List-II). The key concept is dimensional analysis: every physical quantity can be expressed as a product of powers of the fundamental dimensions mass ($M$), length ($L$), time ($T$), and sometimes electric current ($I$), temperature ($\Theta$), etc.

The four quantities are:

1. Rydberg constant ($R_H$): appears in the Rydberg formula for hydrogen spectral lines: $\frac{1}{\lambda} = R_H \Bigl(\frac{1}{n_1^2} - \frac{1}{n_2^2}\Bigr).$ Since $1/\lambda$ has dimension of inverse length, $R_H$ must have dimension $[L^{-1}]$.
2. Planck’s constant ($h$): relates energy of a photon to its frequency: $E = h\nu.$ Energy has dimension $[M\,L^2\,T^{-2}]$, frequency has dimension $[T^{-1}]$, so $h$ has dimension $[M\,L^2\,T^{-1}]$.
3. Magnetic field energy density ($u_B$): energy stored per unit volume in a magnetic field is $u_B = \frac{B^2}{2\mu_0}.$ Energy density has dimension $[M\,L^{-1}\,T^{-2}]$.
4. Coefficient of viscosity ($\eta$): defined by Newton’s law of viscosity: $F = \eta A \frac{dv}{dx}.$ Force has dimension $[M\,L\,T^{-2}]$, area $[L^2]$, velocity gradient $[T^{-1}]$, so $\eta$ has dimension $[M\,L^{-1}\,T^{-1}]$.

Hence the correct matching is
(a) → (iii), (b) → (ii), (c) → (iv), (d) → (i).
This corresponds to option B.

1. Confusing energy density with pressure: both have the same dimension $[M\,L^{-1}\,T^{-2}]$, but the question explicitly asks for magnetic field energy density, not pressure. 2. Misreading the list: students sometimes swap (c) and (d) because both involve $kg\,m^{-1}\,s^{-n}$. 3. Planck’s constant dimension: a frequent error is to write $[M\,L^2\,T^{-2}]$ (energy) instead of $[M\,L^2\,T^{-1}]$. 4. Rydberg constant: it is easy to forget that $R_H$ has units of inverse length, not inverse time or energy.

Exam Tip: Always write down the defining equation first, then substitute dimensions step by step to avoid sign or power errors.