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Dimensions of physical quantities, dimensional analysis and applications: NEET MCQs with answers

50 questions on this topic of Physics and Measurement (NEET 2027 syllabus). Here are 12; tap one to see its answer and explanation.

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easy

1. The dimensional formula of the coefficient of thermal conductivity (K) is:

  1. [M L² T⁻² K⁻¹]
  2. [M L² T⁻³ K⁻¹]
  3. [M L T⁻² K⁻¹]
  4. [M L T⁻³ K⁻¹]
Show answer and explanation

Answer: (D) [M L T⁻³ K⁻¹]

From the rate of heat flow equation, dQ/dt = K A (ΔT/Δx), we have K = (dQ/dt) · (Δx / (A · ΔT)). Substituting dimensions gives [M L² T⁻³] · [L] / ([L²] · [K]) = [M L T⁻³ K⁻¹].

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2. If density (ρ), velocity (v), and frequency (ν) are chosen as the fundamental quantities in a system of units, the dimensional formula for surface tension (S) is:

  1. [ρ v³ ν⁻¹]
  2. [ρ v² ν⁻²]
  3. [ρ v³ ν]
  4. [ρ² v ν⁻¹]
Show answer and explanation

Answer: (A) [ρ v³ ν⁻¹]

Let S = k ρᵃ vᵇ νᶜ. Dimensions of surface tension [S] = [M T⁻²], density [ρ] = [M L⁻³], velocity [v] = [L T⁻¹], and frequency [ν] = [T⁻¹]. Equating dimensions: [M T⁻²] = [M L⁻³]ᵃ [L T⁻¹]ᵇ [T⁻¹]ᶜ = [Mᵃ L⁻³ᵃ⁺ᵇ T⁻ᵇ⁻ᶜ]. For M: a = 1. For L: -3a + b = 0 ⇒ b = 3. For T: -b - c = -2 ⇒ -3 - c = -2 ⇒ c = -1. Thus, [S] = [ρ v³ ν⁻¹].

hard

3. A quantity Q is given by Q = √(L / C), where L is self-inductance and C is capacitance of an electrical circuit. The dimensional formula of Q is identical to that of:

  1. Resistance
  2. Inductive reactance per unit time
  3. Capacitance
  4. Electric potential
Show answer and explanation

Answer: (A) Resistance

The quantity √(L / C) has units of Henry / Farad raised to 1/2, which is equivalent to Ohms (electrical resistance). Dimensionally, [L] = [M L² T⁻² A⁻²] and [C] = [M⁻¹ L⁻² T⁴ A²], so [L / C] = [M² L⁴ T⁻⁶ A⁻⁴], and its square root is [M L² T⁻³ A⁻²], which is the dimension of resistance.

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4. In the expression x = (a t² + b) / (c + t), x represents position and t denotes time. The dimensional formula of the ratio a / c is:

  1. [M⁰ L T⁻²]
  2. [M⁰ L T⁻¹]
  3. [M⁰ L T⁻³]
  4. [M⁰ L² T⁻¹]
Show answer and explanation

Answer: (A) [M⁰ L T⁻²]

By the principle of homogeneity, the denominator requires [c] = [t] = [T]. The entire expression gives [x] = [a t²] / [t] = [a t] = [L], so [a] = [L T⁻¹]. Thus, the ratio [a/c] = [L T⁻¹] / [T] = [M⁰ L T⁻²].

easy

5. A hypothetical damping force acting on a sphere of radius r moving with speed v through a fluid is modeled as F = 6π ηᵃ rᵇ vᶜ. If the coefficient of viscosity η has dimensions [M L⁻¹ T⁻¹], then dimensional consistency dictates that:

  1. a = 1, b = 1, c = 1
  2. a = 1, b = 2, c = 1
  3. a = 2, b = 1, c = 1
  4. a = 1, b = 1, c = 2
Show answer and explanation

Answer: (A) a = 1, b = 1, c = 1

Equating dimensions for Stoke's formula: [M L T⁻²] = [M L⁻¹ T⁻¹]ᵃ [L]ᵇ [L T⁻¹]ᶜ = [M]ᵃ [L]^(-a + b + c) [T]^(-a - c). Equating exponents: a = 1; -a - c = -2 ⇒ c = 1; and -a + b + c = 1 ⇒ -1 + b + 1 = 1 ⇒ b = 1.

hard

6. In the relation F = A e^(-B t) + C ln(D x), where F is force, t is time, and x is position, the dimensional formula of (A · D / B) is:

  1. [M L² T⁻²]
  2. [M L⁰ T⁻¹]
  3. [M L T⁻¹]
  4. [M L⁻¹ T⁰]
Show answer and explanation

Answer: (B) [M L⁰ T⁻¹]

From the principle of homogeneity, the argument of the exponential is dimensionless, so [B t] = 1 ⇒ [B] = [T⁻¹]. Also [A] = [F] = [M L T⁻²]. The argument of the logarithm must be dimensionless, so [D x] = 1 ⇒ [D] = [L⁻¹]. Therefore, the dimensions of (A · D / B) are [M L T⁻²] · [L⁻¹] / [T⁻¹] = [M L⁰ T⁻¹].

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7. The torque τ acting on a magnetic dipole of magnetic moment M in a uniform magnetic field B is given by τ = M × B. What is the dimensional formula of magnetic permeability of vacuum (μ₀)?

  1. [M L T⁻² A⁻²]
  2. [M L² T⁻² A⁻¹]
  3. [M L⁻¹ T⁻² A²]
  4. [M L³ T⁻¹ A⁻²]
Show answer and explanation

Answer: (A) [M L T⁻² A⁻²]

Magnetic field B has dimensions [M T⁻² A⁻¹]. From Ampere's law, B = μ₀ I / (2π r), so [μ₀] = [B][L] / [A] = [M T⁻² A⁻¹][L] / [A] = [M L T⁻² A⁻²].

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8. If momentum (p), area (A), and time (T) are taken as fundamental physical quantities, the dimensional formula for energy (E) is:

  1. [p A¹/² T]
  2. [p² A¹/² T⁻¹]
  3. [p A¹/² T⁻¹]
  4. [p A T⁻¹]
Show answer and explanation

Answer: (C) [p A¹/² T⁻¹]

Energy has dimensions [M L² T⁻²]. Momentum is [p] = [M L T⁻¹], so [M] = [p L⁻¹ T]. Area has [A] = [L²], giving [L] = [A¹/²]. Substituting these gives [E] = [p] [L] [T⁻¹] = [p A¹/² T⁻¹].

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9. The viscous force F acting between two parallel liquid layers of contact area A separated by distance dx and having relative velocity dv is given by F = -η A (dv / dx). The dimensions of the coefficient of viscosity η are:

  1. [M L⁻² T⁻²]
  2. [M L⁻¹ T⁻¹]
  3. [M L T⁻¹]
  4. [M L⁻¹ T⁻²]
Show answer and explanation

Answer: (B) [M L⁻¹ T⁻¹]

From the equation, η = F / (A · (dv / dx)). The dimensions are [F] = [M L T⁻²], [A] = [L²], and [dv / dx] = [T⁻¹]. Thus, [η] = [M L T⁻²] / ([L²] [T⁻¹]) = [M L⁻¹ T⁻¹].

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10. Which pair of physical quantities has the same dimensions?

  1. Work and torque
  2. Force and power
  3. Momentum and energy
  4. Pressure and force
Show answer and explanation

Answer: (A) Work and torque

Work (force × distance) and torque (force × perpendicular distance) both have dimensions [M L² T⁻²], though they are different physical quantities.

NCERT Class 11

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11. The rate of flow of a liquid (volume V flowing per unit time t) through a horizontal capillary tube of radius r and length l under a pressure difference P is given by Poiseuille's formula: V/t = (π P r⁴) / (8 η l). The dimensional formula of dynamic viscosity η derived from this equation is:

  1. [M L T⁻¹]
  2. [M L⁻² T⁻¹]
  3. [M L⁻¹ T⁻²]
  4. [M L⁻¹ T⁻¹]
Show answer and explanation

Answer: (D) [M L⁻¹ T⁻¹]

Volume per unit time [V/t] = [L³ T⁻¹]. Pressure [P] = [M L⁻¹ T⁻²]. Rearranging gives [η] = [P r⁴] / ([V/t] l) = ([M L⁻¹ T⁻²][L⁴]) / ([L³ T⁻¹][L]) = [M L⁻¹ T⁻¹].

easy

12. The dimensional formula of magnetic flux is:

  1. [M L² T⁻¹ A⁻²]
  2. [M L T⁻² A⁻¹]
  3. [M L² T⁻² A⁻¹]
  4. [M L⁰ T⁻² A⁻¹]
Show answer and explanation

Answer: (C) [M L² T⁻² A⁻¹]

Magnetic flux Φ = B × A. Since force on a current-carrying wire is F = I L B, [B] = [F / (I L)] = [M L T⁻²] / ([A][L]) = [M T⁻² A⁻¹]. Therefore, [Φ] = [B][A] = [M T⁻² A⁻¹][L²] = [M L² T⁻² A⁻¹].

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