Quick Recap — Dimensional Analysis

  • Core dimensional formulas: force [MLT−2][MLT^{-2}], energy/work/torque [ML2T−2][ML^2T^{-2}], power [ML2T−3][ML^2T^{-3}], pressure/stress/Young's modulus [ML−1T−2][ML^{-1}T^{-2}], momentum/impulse [MLT−1][MLT^{-1}].
  • Useful constants: Planck's constant hh (and angular momentum) [ML2T−1][ML^2T^{-1}]; gravitational constant GG [M−1L3T−2][M^{-1}L^3T^{-2}]; viscosity η\eta [ML−1T−1][ML^{-1}T^{-1}]; surface tension [MT−2][MT^{-2}].
  • Electromagnetic: charge [AT][AT], electric field [MLT−3A−1][MLT^{-3}A^{-1}], magnetic field [MT−2A−1][MT^{-2}A^{-1}]; the time constants RCRC and LR\dfrac{L}{R} both have dimensions [T][T].
  • Dimensionless: strain, refractive index, angle, relative density.
  • Uses: check equations for consistency and deduce how a quantity depends on others (up to a numerical constant).

Worked mini-example. From E=hνE=h\nu, [h]=[E][ν]=ML2T−2T−1=[ML2T−1][h]=\dfrac{[E]}{[\nu]}=\dfrac{ML^2T^{-2}}{T^{-1}}=[ML^2T^{-1}] — the same dimensions as angular momentum.