Every formula, worked example, and free calculator for Ohm's law, watts, amps, volts, wire sizing, heat transfer, wavelength, free fall, gear ratio, projectile motion, and 70+ more electrical and physics calculations — all verified and in one place.
The foundation of every circuit calculation — voltage, current, resistance, and power interrelationships.
Ohm's Law defines how voltage, current, and resistance relate in any DC circuit. Combined with the power formula P = V x I, you get twelve equations from just four variables (V, I, R, P). Master these four core forms and you can solve any basic circuit problem.
V = I x R (Voltage = Current x Resistance)
I = V / R (Current = Voltage / Resistance)
R = V / I (Resistance = Voltage / Current)
P = V x I (Power = Voltage x Current)
P = I^2 x R (Power from current and resistance)
P = V^2 / R (Power from voltage and resistance)
Calculating watts from amps and volts is the most common electrical calculation in residential and commercial electrical work. A standard 120V outlet circuit rated at 15A can deliver a maximum of 1,800 watts (120 x 15). A 240V circuit at 30A delivers 7,200 watts — enough for an electric dryer or water heater.
In AC circuits, power has three components. Real power (watts) does actual work. Reactive power (VAR) is stored and returned by inductors and capacitors. Apparent power (VA) is what the utility measures and bills. Power factor bridges real and apparent power.
Apparent Power (VA) = V x I
Real Power (W) = V x I x cos(phi) = VA x PF
Reactive Power (VAR) = V x I x sin(phi)
Power Factor (PF) = Real Power / Apparent Power = cos(phi)
VA^2 = W^2 + VAR^2 (Power triangle Pythagorean relationship)
| Motor HP | Single Phase 120V | Single Phase 240V | 3-Phase 240V |
|---|---|---|---|
| 0.5 HP | 9.8 A | 4.9 A | 2.8 A |
| 1 HP | 16.0 A | 8.0 A | 4.6 A |
| 2 HP | 24.0 A | 12.0 A | 6.8 A |
| 5 HP | 56.0 A | 28.0 A | 15.2 A |
| 10 HP | 100.0 A | 50.0 A | 28.0 A |
| 15 HP | — | 72.0 A | 42.0 A |
| 20 HP | — | 100.0 A | 54.0 A |
KVA (kilovolt-amperes) is apparent power — what a generator or transformer is rated at. KW (kilowatts) is real power — what actually does work. KWh (kilowatt-hours) is energy consumed over time — what your electricity bill is measured in. KW = KVA x Power Factor. KWh = KW x Hours.
Convert between watts, amps, volts, VA, KVA, and KW across single-phase and three-phase systems.
Converting watts to amps requires knowing the voltage. For DC circuits and single-phase AC: Amps = Watts / Volts. For three-phase AC: Amps = Watts / (Volts x 1.732 x Power Factor). The 1.732 factor is the square root of 3, accounting for the three-phase relationship.
DC / Single-phase: Amps = Watts / Volts
Single-phase AC: Amps = Watts / (Volts x PF)
3-phase AC: Amps = Watts / (Volts x 1.732 x PF)
Example: 1000W, 120V, PF=1.0 = 1000/120 = 8.33 Amps
| KVA | KW (PF=0.8) | Amps @ 208V 3φ | Amps @ 480V 3φ |
|---|---|---|---|
| 5 KVA | 4 KW | 13.9 A | 6.0 A |
| 10 KVA | 8 KW | 27.8 A | 12.0 A |
| 25 KVA | 20 KW | 69.4 A | 30.1 A |
| 50 KVA | 40 KW | 138.8 A | 60.1 A |
| 100 KVA | 80 KW | 277.6 A | 120.3 A |
| 500 KVA | 400 KW | 1,388 A | 601.4 A |
Select the correct wire gauge for any ampacity, distance, and voltage — based on NEC 2023 ampacity tables.
American Wire Gauge (AWG) uses an inverse scale — lower numbers mean thicker wire with higher current capacity. Every 3 AWG steps doubles the cross-sectional area. NEC Table 310.16 defines ampacity for copper conductors in conduit at 75°C.
| AWG | Ampacity (Cu, 75°C) | Diameter (mm) | Resistance (Ω/1000ft) | Typical Use |
|---|---|---|---|---|
| 14 AWG | 15 A | 1.63 | 3.14 | Lighting, 15A branch circuits |
| 12 AWG | 20 A | 2.05 | 1.98 | Outlets, 20A branch circuits |
| 10 AWG | 30 A | 2.59 | 1.24 | Dryers, water heaters |
| 8 AWG | 40 A | 3.26 | 0.778 | Ranges, AC units |
| 6 AWG | 55 A | 4.11 | 0.491 | EV chargers, subpanels |
| 4 AWG | 70 A | 5.19 | 0.308 | Subpanels, large equipment |
| 2 AWG | 95 A | 6.54 | 0.194 | Service entrances |
| 1/0 AWG | 125 A | 8.25 | 0.122 | 200A service feeders |
| 4/0 AWG | 195 A | 11.68 | 0.0766 | Large service entrances |
Voltage drop reduces the voltage delivered to a load at the end of a wire run. The NEC recommends keeping voltage drop below 3% for branch circuits and below 5% combined for feeder plus branch. Longer runs and higher currents demand thicker wire.
VD = 2 x K x I x L / CM
K = 12.9 (copper) or 21.2 (aluminum) at 75 degrees C
I = current in amps, L = one-way length in feet, CM = circular mils of wire
% Voltage Drop = (VD / Source voltage) x 100
Simplified: VD (volts) = 2 x R_per_foot x I x L
Heat capacity, heat transfer, thermal resistance, thermal expansion, and heat index — the physics of thermal energy.
Specific heat capacity (c) is the energy needed to raise 1 kg of a substance by 1 Kelvin. Water has one of the highest specific heats at 4,186 J/kg·K, which is why it's such an effective coolant. Metals have much lower values, which is why they heat up quickly.
Q = m x c x delta-T
Q = heat energy (Joules), m = mass (kg), c = specific heat (J/kg*K)
delta-T = temperature change (Kelvin or Celsius change)
Example: Heat 5 kg of water by 80 degrees C = 5 x 4186 x 80 = 1,674,400 J = 1,674 kJ
Heat transfers through three mechanisms: conduction (through a solid), convection (through a fluid), and radiation (electromagnetic waves). In electrical systems, conduction is most relevant for thermal resistance calculations in components, PCBs, and heat sinks.
Q/t = k x A x delta-T / d
k = thermal conductivity (W/m*K), A = cross-sectional area (m^2)
d = thickness (m), delta-T = temperature difference (K)
Thermal Resistance R_th = d / (k x A) in K/W (like electrical resistance)
Thermal resistance (Rθ) describes how much a component's temperature rises per watt of power dissipated. Junction-to-ambient thermal resistance RθJA is the key parameter for heatsink selection. If a transistor dissipates 10W with RθJA = 8°C/W, it runs 80°C above ambient.
| Material | Specific Heat (J/kg·K) | Common Use |
|---|---|---|
| Water | 4,186 | Cooling systems, boilers |
| Air (dry) | 1,005 | HVAC, airflow calculations |
| Aluminum | 897 | Heat sinks, busbars |
| Copper | 385 | Wire, conductors |
| Iron/Steel | 490 | Motor cores, structural |
| Concrete | 880 | Thermal mass, slab heating |
| Oil (engine) | 1,900 | Transformer cooling |
Newton's laws, velocity, acceleration, friction, torque, and kinematic equations for mechanical and electrical engineering problems.
The five kinematic equations relate displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t) for uniform acceleration. Any problem with three known values can find the remaining two using this system.
v = u + a*t (final velocity)
s = u*t + 0.5*a*t^2 (displacement)
v^2 = u^2 + 2*a*s (velocity-displacement)
s = (u + v)/2 * t (average velocity method)
s = v*t - 0.5*a*t^2 (final velocity form)
Force equals mass times acceleration: F = m x a. The net force acting on an object equals the vector sum of all forces. For friction problems, the friction force opposes motion and equals the friction coefficient multiplied by the normal force: F_friction = μ x N.
| Surface Pair | Static μ | Kinetic μ |
|---|---|---|
| Steel on steel (dry) | 0.74 | 0.57 |
| Steel on steel (lubricated) | 0.15 | 0.10 |
| Rubber on concrete (dry) | 0.90 | 0.80 |
| Rubber on wet concrete | 0.30 | 0.25 |
| Wood on wood | 0.40 | 0.20 |
| Teflon on Teflon | 0.04 | 0.04 |
| Ice on ice | 0.10 | 0.03 |
Velocity, time, distance, height, air resistance — all four solve-for modes plus 9-planet gravitational constants.
Free fall assumes no air resistance (vacuum conditions). In real-world applications, air resistance opposes the fall and reduces acceleration below g. The four key free fall calculations are: velocity from time, time from distance, distance from time, and height from velocity.
Velocity: v = g * t
Distance: d = 0.5 * g * t^2
Time: t = sqrt(2d / g)
Velocity from height: v = sqrt(2 * g * h)
g = 9.81 m/s^2 on Earth (9.807 exact per NIST)
When air resistance is included, the drag force equals 0.5 x C_d x rho x A x v², where C_d is the drag coefficient, rho is air density (1.225 kg/m³ at sea level), A is the cross-sectional area, and v is velocity. Terminal velocity occurs when drag equals gravitational force: v_terminal = sqrt(2mg / (C_d x rho x A)).
| Body | g (m/s²) | vs. Earth | Fall from 10m — Time |
|---|---|---|---|
| Earth | 9.81 | 1.00x | 1.43 s |
| Moon | 1.62 | 0.17x | 3.51 s |
| Mars | 3.72 | 0.38x | 2.32 s |
| Jupiter | 24.79 | 2.53x | 0.90 s |
| Saturn | 10.44 | 1.06x | 1.38 s |
| Venus | 8.87 | 0.90x | 1.50 s |
| Neptune | 11.15 | 1.14x | 1.34 s |
| Sun | 274.0 | 27.9x | 0.27 s |
Horizontal range, maximum height, time of flight, and trajectory for any launch angle and initial velocity.
Projectile motion combines uniform horizontal motion (constant velocity, no acceleration) with vertical free fall (constant acceleration g downward). The two components are completely independent — horizontal velocity never changes, vertical velocity changes at rate g.
Horizontal velocity: vx = v0 * cos(theta) [constant throughout]
Vertical velocity: vy = v0 * sin(theta) - g * t
Horizontal position: x = v0 * cos(theta) * t
Vertical position: y = v0 * sin(theta) * t - 0.5 * g * t^2
Time of flight: T = 2 * v0 * sin(theta) / g
Maximum height: H = (v0 * sin(theta))^2 / (2 * g)
Range: R = v0^2 * sin(2 * theta) / g
Maximum range at theta = 45 degrees
For an object launched horizontally (theta = 0°) from height h, the time of flight is t = sqrt(2h/g), horizontal range is R = v0 x t, and impact velocity combines both components: v_impact = sqrt(vx² + vy²).
Wavelength-to-frequency conversion, energy-to-wavelength, wave speed, and the electromagnetic spectrum reference.
All waves obey the fundamental relationship between speed, frequency, and wavelength. For electromagnetic waves in vacuum (light, radio, X-rays), the speed is always c = 299,792,458 m/s. For sound waves, speed depends on the medium (343 m/s in air at 20°C).
v = f * lambda (wave speed = frequency x wavelength)
f = v / lambda (frequency from speed and wavelength)
lambda = v / f (wavelength from speed and frequency)
For EM waves: lambda = c / f (c = 3 x 10^8 m/s)
Photon energy: E = h * f = h * c / lambda (h = 6.626 x 10^-34 J*s)
Wavenumber: k = 2*pi / lambda (radians per meter)
| Wave Type | Wavelength | Frequency | Energy/Photon |
|---|---|---|---|
| AM Radio | 100 m – 1 km | 300 kHz – 3 MHz | ~1.2 – 12 μeV |
| FM Radio / WiFi | 0.1 – 10 m | 30 MHz – 3 GHz | 0.12 – 12 meV |
| Microwave / 5G | 1 mm – 10 cm | 3 – 300 GHz | 12 – 1,200 meV |
| Infrared | 700 nm – 1 mm | 300 GHz – 430 THz | 1.2 meV – 1.7 eV |
| Visible Light | 380 – 700 nm | 430 – 790 THz | 1.7 – 3.3 eV |
| Ultraviolet | 10 – 380 nm | 790 THz – 30 PHz | 3.3 – 124 eV |
| X-rays | 0.01 – 10 nm | 30 PHz – 30 EHz | 124 eV – 124 keV |
Gear ratios, speed conversions, engine displacement, flow rate, and fluid mechanics for mechanical and automotive engineering.
A gear ratio describes how rotational speed and torque are transformed between a drive gear and a driven gear. A higher ratio (numerically larger) multiplies torque but reduces speed. A lower ratio produces higher output speed with reduced torque — fundamental to transmission design.
Gear Ratio = Driven Teeth / Drive Teeth = Input RPM / Output RPM
Output RPM = Input RPM / Gear Ratio
Output Torque = Input Torque x Gear Ratio x Efficiency
Gear Speed (surface): v = pi x d x N / 60 (d=diameter m, N=RPM)
Overall ratio (compound): GR_total = GR1 x GR2 x GR3 ...
Engine displacement is the total volume swept by all pistons during one full revolution. It directly determines air-fuel mixture volume and is one of the primary factors in power output.
Displacement = (pi / 4) x Bore^2 x Stroke x Cylinders
Bore and Stroke in mm, result in cm^3 (divide by 1000 for liters)
Example: 4-cyl, 86mm bore, 86mm stroke = (pi/4) x 86^2 x 86 x 4 = 1,999 cc (2.0L)
Watts to lumens, watt-hours to amp-hours, joules to volts, foot-pounds, and energy unit conversions for electrical and physics work.
Lumens measure the total visible light output of a source. Watts measure power consumption. The ratio (luminous efficacy) varies enormously by technology: incandescent bulbs produce about 15 lumens/watt, LEDs produce 80–150 lumens/watt. To replace a 60W incandescent, look for an LED rated at 800 lumens (60 x 13 lm/W).
Lumens = Watts x Luminous Efficacy (lm/W)
Lux = Lumens / Area (m^2) [illuminance on a surface]
Candela = Lumens / (4 * pi) [point source, all directions]
Foot-candles = Lux / 10.764 [1 fc = 10.764 lux]
Typical efficacy: LED 80-150, CFL 50-70, Halogen 15-25, Incandescent 10-15 lm/W
Battery capacity is often listed in amp-hours (Ah) or milliamp-hours (mAh). Watt-hours gives the energy stored regardless of voltage. To convert: Wh = Ah x Voltage. A 12V 100Ah car battery stores 1,200 Wh (1.2 kWh). A phone battery rated 4,000 mAh at 3.85V stores 15.4 Wh.
Concentration, enthalpy, combustion, gas laws, density, and molecular chemistry calculations used in electrical and chemical engineering.
Gay-Lussac's Law relates pressure and temperature at constant volume: P1/T1 = P2/T2 (temperatures in Kelvin). This is critical for pressurized electrical equipment like SF6 circuit breakers, where internal gas pressure must be monitored against temperature.
PV = nRT
P = pressure (Pa), V = volume (m^3), n = moles, R = 8.314 J/mol*K, T = Kelvin
Combined Gas Law: P1*V1/T1 = P2*V2/T2
Gay-Lussac's (constant V): P1/T1 = P2/T2
Boyle's Law (constant T): P1*V1 = P2*V2
Radioactive decay follows first-order kinetics: N(t) = N0 x e^(-lambda x t), where lambda = ln(2) / t_half. Carbon-14 has a half-life of 5,730 years. After 10 half-lives, only 1/1024 of the original material remains. Carbon dating is accurate for materials up to approximately 50,000 years old.
All formulas and reference data on this guide are sourced from current standards and authoritative references: