Formula & Dimensional Analysis PhysicsIQ

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#	Physical Quantity	Formula Basis	Dimensional Formula	Topic
1	Area	l × b	[L²]	Mechanics
2	Volume	l³	[L³]	Mechanics
3	Velocity / Speed	s/t	[LT⁻¹]	Mechanics
4	Acceleration	v/t	[LT⁻²]	Mechanics
5	Momentum	mv	[MLT⁻¹]	Mechanics
6	Force	ma	[MLT⁻²]	Mechanics
7	Impulse	F·Δt	[MLT⁻¹]	Mechanics
8	Work / Energy	F·s	[ML²T⁻²]	Mechanics
9	Power	W/t	[ML²T⁻³]	Mechanics
10	Pressure / Stress	F/A	[ML⁻¹T⁻²]	Mechanics
11	Density	m/V	[ML⁻³]	Mechanics
12	Strain	ΔL/L (ratio)	[M⁰L⁰T⁰]	Dimensionless
13	Surface Tension	F/l	[MT⁻²]	Mechanics
14	Coefficient of Viscosity	stress/velocity gradient	[ML⁻¹T⁻¹]	Mechanics
15	Gravitational Constant G	Fr²/m²	[M⁻¹L³T⁻²]	Mechanics
16	Spring Constant k	F/x	[MT⁻²]	Mechanics
17	Angular Momentum	r × p	[ML²T⁻¹]	Mechanics
18	Torque	r × F	[ML²T⁻²]	Mechanics
19	Frequency	1/t	[T⁻¹]	Mechanics
20	Specific Heat Capacity	Q/(m·ΔT)	[L²T⁻²K⁻¹]	Thermal
21	Thermal Conductivity	Q·l/(A·ΔT·t)	[MLT⁻³K⁻¹]	Thermal
22	Boltzmann Constant k_B	E/T	[ML²T⁻²K⁻¹]	Thermal
23	Planck's Constant h	E/f	[ML²T⁻¹]	Modern Physics
24	Electric Field E	F/q	[MLT⁻³A⁻¹]	Electromagnetism
25	Potential Difference V	W/q	[ML²T⁻³A⁻¹]	Electromagnetism
26	Resistance R	V/I	[ML²T⁻³A⁻²]	Electromagnetism
27	Capacitance C	Q/V	[M⁻¹L⁻²T⁴A²]	Electromagnetism
28	Resistivity ρ	RA/l	[ML³T⁻³A⁻²]	Electromagnetism
29	Magnetic Field B	F/(qv)	[MT⁻²A⁻¹]	Electromagnetism
30	Permittivity ε₀	from Coulomb's law	[M⁻¹L⁻³T⁴A²]	Electromagnetism
31	Permeability μ₀	from Biot-Savart	[MLT⁻²A⁻²]	Electromagnetism
32	Refractive Index	c/v (ratio)	[M⁰L⁰T⁰]	Dimensionless
33	Angle (radian)	arc/radius (ratio)	[M⁰L⁰T⁰]	Dimensionless

3 Power Applications of Dimensional Analysis

① Checking Correctness of an Equation

The Principle of Homogeneity: LHS and RHS must have the same dimensions for an equation to be physically valid.

s = ut + ½at²

[s] = [L]

[ut] = [LT⁻¹][T] = [L] ✓

[½at²] = [LT⁻²][T²] = [L] ✓

LHS = RHS dimensionally → Equation is dimensionally correct

❌

Critical Limitation

Dimensionally correct ≠ Physically correct. The equation s = ut + 2at² is also dimensionally correct but physically wrong (coefficient should be ½). Dimensions cannot detect numerical constants.

② Deriving Relationships

Assume T ∝ lᵃgᵇ and find a, b using dimensional matching.

Write: [T] = [L]ᵃ[LT⁻²]ᵇ

[T¹] = [L^(a+b)] [T^(−2b)]

Compare T: −2b = 1 → b = −½

Compare L: a + b = 0 → a = ½

∴ T ∝ √(l/g)

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Strategy Tip

Dimensional analysis gives the form but NOT the dimensionless constant (2π in this case). For JEE, always remember: this method cannot give you 1, 2, π, ½ etc.

③ Converting Units Between Systems

If a quantity has dimension [MᵃLᵇTᶜ], then:
n₂ = n₁ × (M₁/M₂)ᵃ × (L₁/L₂)ᵇ × (T₁/T₂)ᶜ

Example: Convert viscosity from CGS to SI

Viscosity = [ML⁻¹T⁻¹]

M: 1 g → 10⁻³ kg → factor: 10⁻³
L: 1 cm → 10⁻² m → factor: (10⁻²)⁻¹ = 10²
T: 1 s → 1 s → factor: 1

1 poise = 1 g cm⁻¹ s⁻¹ = 10⁻³ × 10² × 1 = 0.1 kg m⁻¹ s⁻¹ = 0.1 Pa·s

📐 Derivation 1 — Time Period of Simple Pendulum

Assume T depends on length l, acceleration due to gravity g, and mass m.

Assume: T = k · mˣ · lʸ · gᶻ

Dimensions: [T] = [M]ˣ[L]ʸ[LT⁻²]ᶻ

[T¹] = [Mˣ · L^(y+z) · T^(−2z)]

Compare M: x = 0 (mass doesn't matter!)

Compare T: −2z = 1 → z = −½

Compare L: y + z = 0 → y = ½

T = k · √(l/g) → T ∝ √(l/g)

Note: Dimensional analysis gives T ∝ √(l/g). It cannot give k = 2π. That requires experiment or full physics.

📐 Derivation 2 — Speed of Transverse Wave on String

Assume v depends on tension T and linear mass density μ (mass per unit length).

v = k · Tᵃ · μᵇ

[T] = [MLT⁻²], [μ] = [ML⁻¹]

[LT⁻¹] = [MLT⁻²]ᵃ · [ML⁻¹]ᵇ

[LT⁻¹] = [M^(a+b) · L^(a−b) · T^(−2a)]

M: a + b = 0, L: a − b = 1, T: −2a = −1

a = ½, b = −½

v ∝ √(T/μ)

📐 Derivation 3 — Stokes' Law for Viscous Drag

Force F depends on viscosity η, radius r, and velocity v.

F = k · ηᵃ · rᵇ · vᶜ

[η] = [ML⁻¹T⁻¹], [r] = [L], [v] = [LT⁻¹]

[MLT⁻²] = [ML⁻¹T⁻¹]ᵃ[L]ᵇ[LT⁻¹]ᶜ

M: a=1, T: −a−c=−2 → c=1, L: −a+b+c=1 → b=1

F = k · η · r · v → Stokes' Law: F = 6πηrv

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Exam Insight — JEE Main

JEE Main PYQs consistently test torque, viscosity, μ₀, ε₀, spring constant, thermal conductivity, and matching-type dimensional questions. These tricks make you 3× faster in elimination.

Arguments Must Be Dimensionless

If quantity is inside sin(x), cos(x), eˣ, log(x) — x MUST be dimensionless. Use this to find missing dimensions instantly.

Ratio = Dimensionless

Any ratio of quantities with the same dimension is dimensionless. Strain, refractive index, angle, efficiency — check if it's a ratio first.

Eliminate Options Fast

If options differ dimensionally, eliminate without solving algebra. Find the dimension of the target, match options. Works 40% of the time in JEE MCQs.

Addition Law

In x + y + z, ALL terms must have the same dimension. If dimensions differ, the expression is physically invalid regardless of numerical values.

Dimensionally Correct ≠ Physically Complete

s = ut + 2at² is dimensionally correct but wrong. This is the single most tested conceptual trap in assertion-reason questions.

Constants Are Dimensionless

2, π, e, ½, 4π, 6π — pure numbers have no dimension. Dimensional analysis cannot determine them. State this explicitly in derivations for full marks.

Same Dimension, Different Quantities

Torque and Work both have [ML²T⁻²]. Impulse and Momentum both have [MLT⁻¹]. Planck's constant h and angular momentum L both have [ML²T⁻¹]. High-yield JEE trap.

Speed of Light Trick

[c] = [LT⁻¹]. And [1/√(μ₀ε₀)] = [LT⁻¹]. This is why c = 1/√(μ₀ε₀). Dimension-matching reveals physical relationships.

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Strategy Tip — JEE Main Time Saver

For MCQs: Before writing equations, do dimensional elimination on the 4 options. If 2 options are eliminated in 15 seconds, you reduce guessing probability by 50% even if you can't solve the full question.

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⚡ Formula & Dimensional Analysis