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Electricity

Coulomb\'s law calculator with the constant k = 8.99 × 10⁹ N·m²/C², plus a reference covering impedance, DC vs AC, two-phase and three-phase systems, and the formulas that connect them all. The physics anchor for every other calculator on this site. Reviewed by a licensed PE.

Use the Coulomb\'s law calculator

Coulomb\'s law is the foundational equation for the electric force between two point charges, with the constant k = 8.99 × 10⁹ N·m²/C² appearing as the proportionality. Enter two charges, the distance between them, and pick a medium — the calculator returns force, electric field, electric potential, and potential energy.

CALC.006 Coulomb's Law · F · E · V · U · k = 8.99 × 10⁹ N·m²/C²

Charge q₁[q₁]

Charge q₂[q₂]

Charge unit[Q]

Distance r between charges[r]

Distance unit[L]

Medium[εr]

Coulomb's constant in vacuum: k = 8.99 × 10⁹ N·m²/C². In a dielectric medium, the effective k is reduced by factor εr.

Force F between charges

— N

Force direction depends on charge signs: like signs repel, opposite signs attract.

Force direction: —
Effective k (vacuum: 8.99e9): —
Electric field at q₂ position: —
Electric potential at q₂ position: —
Potential energy U: —

FORMULA · F = k · q₁ · q₂ / r² SOURCE · COULOMB 1785 · SI 2019

Figure 1 — Coulomb's Law: electrostatic force between two point charges, with the historic torsion balance inset

Electricity — at a glance

Coulomb constant k: k = 8.99 × 10⁹ N·m²/C² (exactly 8.9875517923 × 10⁹). Equivalent: k = 1/(4π·ε₀).
Coulomb's law: F = k · q₁ · q₂ / r² — force in newtons between two point charges, attractive if opposite, repulsive if same sign.
Impedance: Z = √(R² + X²) in ohms. Combines resistance R (in-phase) and reactance X (90° out-of-phase).
Inductive reactance: X_L = 2π·f·L — increases with frequency and inductance.
Capacitive reactance: X_C = 1/(2π·f·C) — decreases with frequency and capacitance.
DC vs AC: DC = constant direction (battery, solar). AC = sine wave reversing 50/60 Hz (mains, generator).
Father of electricity: No single inventor. Franklin (1750s), Faraday (1830s induction), Maxwell (1860s field equations), Edison and Tesla (commercialisation) all share the title.

The electricity formulas

Three families of formulas cover most practical electrical engineering: the electrostatic relations (Coulomb), the resistive-circuit relations (Ohm), and the AC impedance relations (Z, X). Each underlies a different class of calculator on this site.

Eq. 01 — Coulomb's law (electric force between two point charges) SI · Coulomb 1785

F = \frac{k \cdot q_{1} \cdot q_{2}}{r^{2}} \qquad k = \frac{1}{4 \pi \varepsilon_{0}} \approx 8.99 \times 10^{9} \, \frac{N \cdot m^{2}}{C^{2}}

F: magnitude of electric force, N
q_1, q_2: point charges (signed), C
r: distance between the charges, m
k: Coulomb's constant, N·m²/C²
ε_0: permittivity of free space ≈ 8.854 × 10⁻¹² F/m, F/m

Eq. 02 — Ohm's law (resistive circuit) SI · Ohm 1827

V = I \cdot R \qquad P = V \cdot I = I^{2} \cdot R = \frac{V^{2}}{R}

V: voltage across the element, V
I: current through it, A
R: resistance, Ω
P: electrical power dissipated, W

Eq. 03 — Impedance in an AC circuit SI · Steinmetz 1893

Z = \sqrt{R^{2} + X^{2}} \qquad X_{L} = 2 \pi f L \qquad X_{C} = \frac{1}{2 \pi f C}

Z: impedance magnitude, Ω
R: resistance (in-phase), Ω
X: net reactance (90° component), Ω
X_L: inductive reactance, Ω
X_C: capacitive reactance, Ω
f: frequency, Hz

Worked example: force between two 1 µC charges

Two point charges of +1 µC each, separated by 50 cm in air. What force acts on each charge?

Step	Calculation	Result
Convert charges to SI	1 µC = 1 × 10⁻⁶ C	10⁻⁶ C each
Convert distance to SI	50 cm = 0.5 m	0.5 m
Apply Coulomb\'s law	F = (8.99 × 10⁹) × (10⁻⁶) × (10⁻⁶) / 0.5²	3.6 × 10⁻² N
Air correction (εᵣ ≈ 1.0006)	negligible — divide by 1.0006	~3.6 × 10⁻² N
Result	both charges positive → repulsive	~36 mN, repulsive
Same charges in water (εᵣ ≈ 80)	F / 80	~0.45 mN

The 80× reduction in water explains why dielectric materials are used to insulate capacitor plates — they let you store much more charge at the same voltage by reducing the effective Coulomb force trying to push the charges apart.

Resistance, reactance, and impedance compared

The biggest source of confusion in AC circuits is the difference between three quantities that all have units of ohms but mean different things.

Quantity	Symbol	What it represents	Frequency dependence	Energy behaviour
Resistance	R	Opposition that dissipates energy as heat	None (constant)	P = I²R energy lost to heat
Inductive reactance	X_L	Opposition that stores energy in a magnetic field	Increases with f: X_L = 2πfL	Energy stored, returned each cycle (no net loss)
Capacitive reactance	X_C	Opposition that stores energy in an electric field	Decreases with f: X_C = 1/(2πfC)	Energy stored, returned each cycle (no net loss)
Impedance	Z	Total opposition combining all three above	Generally frequency-dependent	Sum of dissipation and storage; phase angle φ = arctan(X/R)

For pure DC (f = 0): X_L = 0, X_C = ∞ (open), Z = R. For pure AC at high frequency: X_L dominates in inductors, X_C shrinks toward zero in capacitors. This is why a capacitor blocks DC but passes AC, a