MPSC 2026 Prelims Paper-I · Question 53 of 100

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.

• (1)(a)-(iii), (b)-(i), (c)-(ii) — 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.
• (2)(a)-(ii), (b)-(i), (c)-(iii) — 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.
• (4)(a)-(i), (b)-(ii), (c)-(iii) — 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).

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.

The three quantities in this question are chosen so that each belongs to a different branch of physics — electromagnetism, fluid mechanics and quantum theory — which is what makes the current-bearing formula such a giveaway. Beyond dimensions, each is worth knowing in its own right. Permeability of free space, μ₀, appears in the Biot-Savart law and in Ampère's law and has the SI unit henry per metre; its value is close to 4π × 10⁻⁷ H/m, though since the 2019 revision of the SI it is an experimentally determined rather than an exactly defined quantity. The coefficient of viscosity measures a fluid's internal friction, has the SI unit pascal-second and the CGS unit poise, and falls sharply with temperature in liquids while rising with temperature in gases — a contrast that is itself a favourite question. Planck's constant, h = 6.62607015 × 10⁻³⁴ joule-second exactly since the 2019 SI redefinition, is the constant of proportionality between the energy of a photon and its frequency, and its fixed value is now what defines the kilogram.

• Permeability of free space μ₀ has the dimensional formula MLT⁻²I⁻², obtained from F/L = μ₀I₁I₂/2πd. Its SI unit is the henry per metre and its value is close to 4π × 10⁻⁷ H/m, though it has been an experimentally determined quantity since the 2019 revision of the SI.
• Coefficient of viscosity η has the dimensional formula ML⁻¹T⁻¹, from F = ηA(dv/dx). Its SI unit is the pascal-second and its CGS unit the poise, with 1 Pa·s equal to 10 poise. ML⁻¹T⁻¹ is also pressure multiplied by time, which is what the SI unit records.
• Planck's constant h has the dimensional formula ML²T⁻¹, from E = hν. Its value is exactly 6.62607015 × 10⁻³⁴ joule-second under the 2019 SI, and that fixed value is now what defines the kilogram.
• Planck's constant shares its dimensional formula with angular momentum, since mvr gives M × LT⁻¹ × L = ML²T⁻¹ — which is why h is called the quantum of action and why the Bohr model quantises angular momentum in units of h/2π.
• Dimensional analysis cannot supply dimensionless constants, cannot handle trigonometric, exponential or logarithmic terms, and cannot distinguish quantities sharing a formula — work and torque are both ML²T⁻², and energy and moment of force cannot be told apart dimensionally.

• Deriving all three rows before looking at the options. In a three-item matching question one confidently known row usually eliminates two or three choices outright, and here either (b)-(iii) or (c)-(i) settles the question on its own.
• Overlooking that only one dimensional formula contains the current I, and only one listed quantity is electromagnetic. That pairing is fixed by inspection and needs no algebra.
• Assuming a shared dimensional formula means the quantities are the same. Planck's constant and angular momentum are both ML²T⁻¹; work and torque are both ML²T⁻². Dimensions cannot distinguish them.
• Misremembering viscosity as ML⁻¹T⁻² instead of ML⁻¹T⁻¹. The first is pressure; viscosity is pressure multiplied by time, which is why its SI unit is the pascal-second.

Matching questions in MPSC's science section are built to reward elimination rather than exhaustive derivation. The Commission normally offers three or four items per column and four permutations, and constructs the wrong options by rotating or swapping pairs rather than by inventing values — which means every printed dimensional formula is usually a genuine formula belonging to one of the quantities on the page. The consequence for technique is direct: identify the row you can fix with certainty, strike out every option that contradicts it, and only then look at what separates the survivors. Dimensional formulae themselves are asked in two further shapes — a direct 'the dimensional formula of X is', and a pairs question asking which two quantities have the same dimensions, where work and torque, and Planck's constant and angular momentum, are the standard answers.

No directly related past PYQ was found.

• practice — not a real PYQ

  Which of the following pairs of physical quantities have the same dimensional formula ?

  • (a)Force and momentum
  • (b)Planck's constant and angular momentum
  • (c)Pressure and coefficient of viscosity
  • (d)Work and power

  Answer(b) Planck's constant and angular momentum, both ML²T⁻¹. Planck's constant is energy divided by frequency, ML²T⁻² over T⁻¹; angular momentum is mvr, giving M × LT⁻¹ × L. Force is MLT⁻² against momentum's MLT⁻¹; pressure is ML⁻¹T⁻² against viscosity's ML⁻¹T⁻¹; and work is ML²T⁻² against power's ML²T⁻³ — each of the other pairs differs by one power of time.

• practice — not a real PYQ

  The SI unit of the coefficient of viscosity is :

  • (a)newton per square metre
  • (b)pascal-second
  • (c)newton-second
  • (d)poise

  Answer(b) pascal-second. The dimensional formula of viscosity, ML⁻¹T⁻¹, is pressure multiplied by time, which is exactly what the unit records. The newton per square metre is the pascal, the unit of pressure alone; the newton-second is the unit of impulse and momentum; and the poise is the CGS unit of viscosity rather than the SI unit, with one pascal-second equal to ten poise.