Dimensional Formula of Energy

Energy is the capacity to do work, and Work = Force × Displacement:

W = F × d

Dimensions of each quantity: • [F] = [M¹L¹T⁻²] (force) • [d] = [L¹] (displacement)

Substituting: [W] = [M¹L¹T⁻²] × [L¹] [W] = [M¹L¹⁺¹T⁻²] [W] = [M¹L²T⁻²]

Dimensional Formula of Energy = [M¹L²T⁻²]

SI unit verification: 1 Joule = 1 N × 1 m = (1 kg·m·s⁻²) × (1 m) = 1 kg·m²·s⁻² This confirms [M¹L²T⁻²] ✓

Alternative derivation from Kinetic Energy: KE = ½mv² [KE] = [M][LT⁻¹]² = [M][L²T⁻²] = [ML²T⁻²] ✓

All forms of energy share the same dimensional formula [ML²T⁻²]:

Kinetic Energy (KE = ½mv²): [KE] = [M][LT⁻¹]² = [ML²T⁻²] ✓

Potential Energy (PE = mgh): [PE] = [M][LT⁻²][L] = [ML²T⁻²] ✓

Elastic Potential Energy (PE = ½kx²): [½kx²] = [MT⁻²][L²] = [ML²T⁻²] ✓

Electrical Energy (E = QV = CV²): [QV] = [AT][ML²T⁻³A⁻¹] = [ML²T⁻²] ✓

Thermal Energy (Q = mcΔT): [mcΔT] = [M][L²T⁻²K⁻¹][K] = [ML²T⁻²] ✓

This dimensional consistency reflects the Law of Conservation of Energy — energy can change form but its dimensional identity remains the same.

SI System: • Unit: Joule (J) • 1 J = 1 N·m = 1 kg·m²·s⁻² • Named after James Prescott Joule

CGS System: • Unit: Erg • 1 erg = 1 g·cm²·s⁻² • 1 J = 10⁷ erg

Other units of energy: • Calorie (cal): 1 cal = 4.186 J (heat energy) • Kilocalorie (kcal): 1 kcal = 4186 J (food energy) • Kilowatt-hour (kWh): 1 kWh = 3.6 × 10⁶ J (electrical energy) • Electronvolt (eV): 1 eV = 1.6 × 10⁻¹⁹ J (atomic/nuclear physics) • BTU (British Thermal Unit): 1 BTU ≈ 1055 J

Large energy units: • 1 kJ = 10³ J • 1 MJ = 10⁶ J • 1 GJ = 10⁹ J

Energy and Work: • Both have dimensional formula [ML²T⁻²] • Both have SI unit Joule (J) • Energy is capacity to do work; work is energy transferred

Power (P = Energy/Time = Work/Time): [P] = [ML²T⁻²] / [T] = [ML²T⁻³] • SI unit: Watt (W) = J/s

Torque (τ = r × F): [τ] = [L][MLT⁻²] = [ML²T⁻²] • Same dimensional formula as energy! • But torque is a vector, energy is a scalar — they are not the same quantity

Angular Momentum (L = Iω = mvr): [L] = [ML²][T⁻¹] = [ML²T⁻¹] • Different from energy

Planck's Constant h: [h] = [Energy]/[frequency] = [ML²T⁻²]/[T⁻¹] = [ML²T⁻¹]

Example 1: Calculate KE of 2 kg object moving at 5 m/s: KE = ½mv² = ½ × 2 × 25 = 25 J Dimensions: [M][LT⁻¹]² = [ML²T⁻²] = Joule ✓

Example 2: Verify Einstein's mass-energy relation E = mc²: [E] = [M][LT⁻¹]² = [ML²T⁻²] ✓ (same as energy) c = speed of light = 3 × 10⁸ m/s E = mc² → 1 kg converts to = 1 × (3×10⁸)² = 9×10¹⁶ J

Example 3: Verify energy stored in capacitor E = ½CV²: [C] = [M⁻¹L⁻²T⁴A²] (capacitance) [V²] = [M²L⁴T⁻⁶A⁻²] (voltage squared) [CV²] = [M⁻¹L⁻²T⁴A²][M²L⁴T⁻⁶A⁻²] = [ML²T⁻²] ✓

Example 4: Gravitational PE at height h: PE = mgh = 5 × 10 × 20 = 1000 J = 1 kJ [mgh] = [M][LT⁻²][L] = [ML²T⁻²] ✓