Match the quantities in Column - I with their dimensions in the Column - II.
Column - I
(a) Permeability (μ₀)
(b) Coefficient of Viscosity (η)
(c) Planck's Constant (h)
Column - II
(i) ML²T⁻¹
(ii) MLT

Question

Match the quantities in Column - I with their dimensions in the Column - II.
Column - I
(a) Permeability (μ₀)
(b) Coefficient of Viscosity (η)
(c) Planck's Constant (h)
Column - II
(i) ML²T⁻¹
(ii) MLT⁻²I⁻²
(iii) ML⁻¹T⁻¹
(Where M is mass, L is length, T is time and I is the current).

UnlockIAS · Accepted Answer

Correct — option (3): (a)-(ii), (b)-(iii), (c)-(i). Each pairing follows from the defining equation of the quantity, and none of the three needs more than one line of substitution. Permeability, μ₀, is defined through the force between two long parallel current-carrying conductors: F/L = μ₀I₁I₂/2πd. Rearranging, μ₀ has the dimensions of force times length divided by length times current squared — that is [MLT⁻²] divided by [I²], giving MLT⁻²I⁻². So (a) matches (ii). The same result comes from B = μ₀I/2πr, since the magnetic field has dimensions MT⁻²I⁻¹. Coefficient of viscosity, η, is defined through Newton's law of viscous flow: F = ηA(dv/dx), where A is the area of the layer and dv/dx the velocity gradient. Therefore η = F ÷ [A × (dv/dx)] = [MLT⁻²] ÷ ([L²] × [T⁻¹]) = ML⁻¹T⁻¹. So (b) matches (iii). A useful cross-check is that ML⁻¹T⁻¹ is pressure multiplied by time, which is exactly what the SI unit of viscosity, the pascal-second, says. Planck's constant, h, is defined through E = hν, so h = E ÷ ν = [ML²T⁻²] ÷ [T⁻¹] = ML²T⁻¹. So (c) matches (i). Note that this is also the dimensional formula of angular momentum, mvr giving M × LT⁻¹ × L = ML²T⁻¹, which is why h is described as the quantum of action and why angular momentum in the Bohr model is quantised in units of h/2π. The combination (a)-(ii), (b)-(iii), (c)-(i) is printed at option (3). There is a much faster route through this question, and it is worth practising because matching items reward it heavily. Start by asking which pairing can be fixed by inspection rather than by derivation. Of the three dimensional formulae offered, only (ii) contains the current I. Of the three quantities offered, only permeability is electromagnetic — viscosity is a property of fluid flow and Planck's constant belongs to energy and frequency, and neither can possibly involve current. So (a) must go with (ii) without any algebra at all. That single observation kills option (1), which pairs (a) with (iii), and option (4), which pairs (a) with (i). Only options (2) and (3) survive, and they differ solely in how they treat (b) and (c). One line settles it: Planck's constant is energy divided by frequency, ML²T⁻² over T⁻¹, which is ML²T⁻¹, that is (i). Option (2) assigns (c) to (iii) and is therefore out. Option (3) stands. Generally, in a three-item matching question the fastest path is to find the one row you are surest of and eliminate on it, rather than to derive all three and then look for the matching combination. Here either the permeability-current observation or the Planck's-constant derivation is on its own sufficient — each of (b)-(iii) and (c)-(i) appears in exactly one option.

Why the other options are wrong — (a) Option (1) gives permeability the dimensions ML⁻¹T⁻¹, which contain no current at all — impossible for a quantity defined by the magnetic force between two currents. It then hands MLT⁻²I⁻², the only formula containing current, to Planck's constant, which is defined by E = hν and involves no electrical quantity whatever. This is a complete rotation of the correct assignment, and it is disposed of by the single observation that current must appear in the permeability row.  (b) Option (2) gets permeability right and then swaps viscosity with Planck's constant, giving viscosity ML²T⁻¹ and Planck's constant ML⁻¹T⁻¹. Both are wrong by the defining equations: η = F ÷ [A × (dv/dx)] yields ML⁻¹T⁻¹, and h = E ÷ ν yields ML²T⁻¹. This is the serious rival to the correct option, because it survives the current-in-the-permeability-row test, and the single line that separates the two is the Planck's constant derivation.  (d) Option (4) assigns ML²T⁻¹ to permeability — again a formula with no current in it, which cannot describe a quantity defined by the force between two currents — and gives MLT⁻²I⁻² to the coefficient of viscosity, implying that a property of fluid flow depends on electric current. Both are impossible on inspection, without any derivation, which is why this option falls at the same first step as option (1).

Concept: Dimensional analysis expresses every physical quantity as a product of powers of a small set of base quantities — mass M, length L, time T, electric current I, thermodynamic temperature K, amount of substance and luminous intensity. The dimensional formula of a derived quantity is read straight off its defining equation, which is why the method requires almost no memory: if you know that viscosity is defined by F = ηA(dv/dx) and that Planck's constant is defined by E = hν, the formulae follow by substitution. The method has three standard uses. It checks equations for consistency, since every additive term must carry the same dimensions — a fast way to catch an algebraic slip. It converts units between systems. And it can derive the form of a relation up to a dimensionless constant, as in the classical derivation that the period of a simple pendulum must go as the square root of length over acceleration due to gravity. Its limits are equally standard and equally examinable: it cannot supply dimensionless constants such as 2π, cannot handle equations involving trigonometric, exponential or logarithmic functions, and cannot distinguish quantities that share a dimensional formula — work and torque both come out as ML²T⁻², and Planck's constant shares ML²T⁻¹ with angular momentum.

State PCS 2026 Q53 — Science & Technology PYQ with Answer

State PCS 2026 Q53 — Science & Technology PYQ with Answer