1. The 7 Fundamental SI Base Units

All physical quantities in mechanics, thermodynamics, and electromagnetism derive from seven fundamental base units defined by the International System of Units (SI): Length (meter, m, dimension [L]); Mass (kilogram, kg, dimension [M]); Time (second, s, dimension [T]); Electric Current (ampere, A, dimension [I] or [A]); Thermodynamic Temperature (kelvin, K, dimension [K] or [θ]); Amount of Substance (mole, mol, dimension [mol]); and Luminous Intensity (candela, cd, dimension [cd]). Supplementary plane angle (radian, rad) and solid angle (steradian, sr) are dimensionless.

2. Core Mechanics Derived Quantities & Dimensions

High-frequency mechanics dimensions tested across board and entrance exams: Velocity: v = dx/dt -> [M^0 L T^-1] (m/s); Acceleration: a = dv/dt -> [M^0 L T^-2] (m/s^2); Linear Momentum & Impulse: p = mv, I = F·Δt -> [M L T^-1] (kg·m/s or N·s); Force: F = ma -> [M L T^-2] (Newton, N); Work, Energy & Torque: W = F·d, τ = r × F -> [M L^2 T^-2] (Joule, J or N·m); Power: P = W/t -> [M L^2 T^-3] (Watt, W); Pressure & Stress: P = F/A -> [M L^-1 T^-2] (Pascal, Pa or N/m^2); Surface Tension: T = F/l -> [M L^0 T^-2] (N/m); Coefficient of Viscosity (η): from Stokes' Law F = 6πηrv -> [M L^-1 T^-1] (Pa·s or poiseuille).

3. Electromagnetism & Modern Physics Dimensional Formulas

Electromagnetism dimensions derived from fundamental laws: Electric Charge: q = I·t -> [M^0 L^0 T A] (Coulomb, C); Electric Potential & EMF: V = W/q -> [M L^2 T^-3 A^-1] (Volt, V); Electric Resistance: R = V/I -> [M L^2 T^-3 A^-2] (Ohm, Ω); Capacitance: C = q/V -> [M^-1 L^-2 T^4 A^2] (Farad, F); Magnetic Field (B): from F = qvB -> [M L^0 T^-2 A^-1] (Tesla, T); Magnetic Flux (Φ): Φ = B·A -> [M L^2 T^-2 A^-1] (Weber, Wb); Inductance (Self & Mutual, L): from e = -L(dI/dt) -> [M L^2 T^-2 A^-2] (Henry, H).

4. Universal Physical Constants: Values, Units & Dimensions

Universal Gravitational Constant (G): 6.674 × 10^-11 N·m^2/kg^2 -> [M^-1 L^3 T^-2]; Planck's Constant (h): 6.626 × 10^-34 J·s -> [M L^2 T^-1] (same dimension as angular momentum); Permittivity of Free Space (ε0): 8.854 × 10^-12 F/m -> [M^-1 L^-3 T^4 A^2]; Permeability of Free Space (μ0): 4π × 10^-7 T·m/A -> [M L T^-2 A^-2]; Speed of Light in Vacuum (c): c = 1/√(μ0 ε0) = 3.0 × 10^8 m/s -> [M^0 L T^-1]; Boltzmann Constant (kB): 1.381 × 10^-23 J/K -> [M L^2 T^-2 K^-1]; Universal Gas Constant (R): 8.314 J/(mol·K) -> [M L^2 T^-2 K^-1 mol^-1].

5. Principle of Homogeneity & Dimensional Analysis Limitations

The Principle of Homogeneity states that in any physically valid equation, the dimensions of all additive terms on both sides of the equals sign must be identical. If equation is y = at + bt^2, dimensions of y, at, and bt^2 must each be identical. Limitations of dimensional analysis: (1) Cannot determine dimensionless proportionality constants (e.g., the 1/2 in s = 1/2 at^2); (2) Fails if a physical quantity depends on more than 3 mechanical variables (M, L, T); (3) Cannot handle trigonometric, exponential, or logarithmic functions directly since their arguments must be dimensionless.

6. Teacher's Exam Hall Hack: Option Elimination via Dimensional Check

When faced with complicated multi-variable algebraic expressions in JEE Main or NEET physics questions (such as finding time period, velocity, or resonant frequency), do not spend 3 minutes deriving the formula from scratch. Calculate the dimensional formula of the requested quantity and check the dimensions of the four multiple-choice options. Incorrect options frequently fail dimensional homogeneity, allowing you to identify the correct option in under 30 seconds.