Coulomb Force Calculator

What is Coulomb's Law?

Coulomb's law quantifies the electric force between two charged objects. Like charges repel, opposite charges attract, and the force strength depends on both the charge magnitudes and the distance between them. This fundamental law governs everything from atomic structure (electrons bound to nuclei) to static cling (socks sticking together from the dryer) to lightning (charge separation in storm clouds).

Imagine two small spheres, each carrying a charge of 1 microcoulomb (1 μC = 10⁻⁶ C), separated by 10 centimeters. Coulomb's law gives F = k × |q₁q₂|/r² = (8.99×10⁹) × (10⁻⁶ × 10⁻⁶)/(0.10)² = 8.99×10⁹ × 10⁻¹²/0.01 = 0.899 N — about the weight of a small apple. Double the distance to 20 cm and the force drops to 0.225 N (one quarter). Double one charge to 2 μC and the force doubles to 1.798 N.

Charles-Augustin de Coulomb discovered this inverse-square relationship in 1785 using a torsion balance — a delicate instrument that measured tiny forces by the twist of a thin wire. The same mathematical form as Newton's gravity (also inverse-square) wasn't coincidental; both forces spread outward spherically from a point source. Coulomb's law is the foundation of electrostatics, enabling calculation of electric fields, potentials, and the behavior of charged particles in everything from particle accelerators to inkjet printers.

How it Works: Formulas Explained

Coulomb's law states F = k × |q₁q₂|/r², where F is the force in newtons, k is Coulomb's constant (8.9875517923×10⁹ N·m²/C², often rounded to 8.99×10⁹ or 9×10⁹), q₁ and q₂ are the charges in coulombs, and r is the separation distance in meters. The force acts along the line connecting the two charges. Like charges (both positive or both negative) produce positive F — repulsion. Opposite charges produce negative F — attraction.

The inverse-square dependence means force drops rapidly with distance. At distance r, force is F. At 2r, force is F/4. At 3r, force is F/9. At 10r, force is F/100. This steep drop explains why you don't feel electric forces from distant charged objects — only nearby charges exert significant force. It also means atomic electrons experience enormous forces from their nuclei (distance ~10⁻¹⁰ m) but negligible forces from neighboring atoms.

Coulomb's constant k = 1/(4πε₀), where ε₀ (epsilon-naught) is the permittivity of free space: ε₀ = 8.854×10⁻¹² C²/(N·m²). In materials other than vacuum, replace ε₀ with ε = κε₀, where κ (kappa) is the dielectric constant. Water has κ ≈ 80, meaning electric forces in water are 80 times weaker than in vacuum — crucial for understanding ionic bonding in aqueous solutions.

Working through a complete example: An electron (q = -1.602×10⁻¹⁹ C) orbits a proton (q = +1.602×10⁻¹⁹ C) at distance r = 5.29×10⁻¹¹ m (Bohr radius). Force F = k × |q₁q₂|/r² = (8.99×10⁹) × (1.602×10⁻¹⁹)²/(5.29×10⁻¹¹)² = 8.99×10⁹ × 2.566×10⁻³⁸/2.80×10⁻²¹ = 8.23×10⁻⁸ N. This tiny force (by everyday standards) keeps the electron bound to the proton, forming a hydrogen atom.

Step-by-Step Guide

1. Identify the charges and their signs. Determine q₁ and q₂ in coulombs. Common values: electron/proton charge e = 1.602×10⁻¹⁹ C, 1 μC = 10⁻⁶ C, 1 nC = 10⁻⁹ C, 1 pC = 10⁻¹² C. Note the signs: like signs mean repulsion, opposite signs mean attraction. The magnitude calculation uses absolute values; direction comes from the signs.
2. Measure the separation distance in meters. Convert all distances to meters: 1 cm = 0.01 m, 1 mm = 0.001 m, 1 nm = 10⁻⁹ m. Example: Two charges 15 cm apart have r = 0.15 m. For atomic-scale problems, distances are often in angstroms (1 Å = 10⁻¹⁰ m) or nanometers. Accurate distance is critical — force depends on r², so a 10% distance error gives 20% force error.
3. Choose the appropriate value of k. For vacuum or air, use k = 8.99×10⁹ N·m²/C². For quick estimates, k ≈ 9×10⁹ works well. In other materials, use k_material = k_vacuum/κ, where κ is the dielectric constant. In water (κ = 80), effective k = 8.99×10⁹/80 = 1.12×10⁸ N·m²/C².
4. Calculate the force magnitude. F = k × |q₁| × |q₂| / r². Example: q₁ = +2 μC, q₂ = -3 μC, r = 0.05 m. F = (8.99×10⁹) × (2×10⁻⁶) × (3×10⁻⁶) / (0.05)² = 8.99×10⁹ × 6×10⁻¹² / 0.0025 = 0.0539 / 0.0025 = 21.6 N. The charges attract with 21.6 newtons of force — about 2.2 kg of weight.
5. Determine the force direction. Like charges repel (force pushes them apart). Opposite charges attract (force pulls them together). For multiple charges, use vector addition. Draw force vectors on each charge: repulsion points away from the other charge, attraction points toward it. In 2D or 3D problems, resolve forces into components.
6. Apply superposition for multiple charges. When more than two charges are present, calculate the force from each charge separately, then add the force vectors. Force from charge A on B is independent of charge C's presence. For three charges in a line, find F_AB and F_CB separately, then add (accounting for direction). For 2D arrangements, use component method or law of cosines.

Real-World Examples

Example 1: Static electricity shock. Walking on carpet builds up ~10,000 V on your body, corresponding to about 10 nC of charge. Touching a doorknob (grounded) at 1 mm distance: the spark jumps when electric field exceeds air's breakdown (~3×10⁶ V/m). Force between your finger and doorknob just before the spark: approximate as point charge q₁ = 10 nC and induced opposite charge q₂ ≈ -10 nC at r = 0.001 m. F = 9×10⁹ × (10⁻⁸)²/(10⁻³)² = 9×10⁹ × 10⁻¹⁶/10⁻⁶ = 0.9 N — nearly 100 grams of attractive force you feel as the spark approaches.

Example 2: Ionic bonding in salt. Sodium chloride (NaCl) consists of Na⁺ and Cl⁻ ions. Charge on each is ±e = ±1.602×10⁻¹⁹ C. Equilibrium separation is about 0.236 nm = 2.36×10⁻¹⁰ m. Attractive force: F = 8.99×10⁹ × (1.602×10⁻¹⁹)²/(2.36×10⁻¹⁰)² = 8.99×10⁹ × 2.566×10⁻³⁸/5.57×10⁻²⁰ = 4.14×10⁻⁹ N. This force, multiplied by Avogadro's number of ions, gives the macroscopic strength of salt crystals. The lattice energy holding salt together originates from these Coulomb attractions.

Example 3: Electrostatic precipitator. Industrial smokestacks use high voltage (~50,000 V) to charge dust particles. A particle with charge q = 100e = 1.6×10⁻¹⁷ C near a collecting plate at 10 cm experiences force from the electric field E = V/d = 50,000/0.1 = 500,000 V/m. Force F = qE = 1.6×10⁻¹⁷ × 5×10⁵ = 8×10⁻¹² N. Tiny as this seems, it dominates over gravity for micron-sized particles, pulling them to the plate and preventing air pollution.

Example 4: Rutherford scattering. Alpha particles (q = +2e) fired at gold nuclei (q = +79e) experience strong repulsion. At closest approach of 10⁻¹⁴ m (nuclear scale): F = 8.99×10⁹ × (2×1.6×10⁻¹⁹) × (79×1.6×10⁻¹⁹)/(10⁻¹⁴)² = 8.99×10⁹ × 4.04×10⁻³⁶/10⁻²⁸ = 3.63 N. This enormous force at subatomic scales (equivalent to lifting 360 grams with a single particle!) deflects alpha particles, revealing the atomic nucleus.

Example 5: Millikan oil drop experiment. Robert Millikan measured the electron charge by balancing electric force against gravity on charged oil droplets. A droplet of mass m = 10⁻¹⁵ kg (weight mg = 10⁻¹⁴ N) with charge q = 3e = 4.8×10⁻¹⁹ C requires electric field E = mg/q = 10⁻¹⁴/(4.8×10⁻¹⁹) = 20,800 V/m to levitate. Between plates 1 cm apart, this needs voltage V = Ed = 208 V. By measuring the voltage needed to suspend droplets, Millikan determined q must be an integer multiple of e = 1.602×10⁻¹⁹ C.

Common Mistakes to Avoid

Forgetting to convert units to SI. Charges must be in coulombs, distances in meters. A common error: using microcoulombs directly (q = 5 instead of 5×10⁻⁶ C) or centimeters for distance (r = 10 instead of 0.10 m). This produces answers wrong by factors of 10⁶ or 10⁴. Always write charges in scientific notation with coulombs: 5 μC = 5×10⁻⁶ C, 100 nC = 1×10⁻⁷ C.

Misapplying the inverse-square relationship. Doubling distance reduces force to 1/4, not 1/2. Tripling distance gives 1/9, not 1/3. Conversely, halving distance increases force 4×, not 2×. This steep dependence surprises many students. If your calculation shows force increasing when distance increases, you've inverted the formula. Check: F should be proportional to 1/r², not r².

Ignoring vector nature in multi-charge problems. Force is a vector with both magnitude and direction. With three or more charges, you can't simply add force magnitudes — you must add force vectors. Two equal forces at 120° angle produce a resultant equal to either force, not twice the force. Draw clear diagrams, label directions, and use component addition (F_x and F_y) for 2D problems.

Confusing Coulomb's law with electric field. Coulomb's law gives force between two specific charges: F = kq₁q₂/r². Electric field is force per unit charge: E = F/q = kQ/r² (field from a single charge Q). Force requires two charges; field exists around a single charge. To find force on charge q in an existing field E, use F = qE. Don't double-count by using both formulas simultaneously.

Pro Tips

Use symmetry to simplify multi-charge problems. For charges arranged symmetrically (equilateral triangle, square, ring), many force components cancel. In an equilateral triangle with identical charges at each corner, the net force on any charge points radially outward. Horizontal components from the other two charges cancel; only vertical components add. Recognizing symmetry cuts calculation work dramatically and reduces errors.

Estimate before calculating. Coulomb's law spans enormous scales — from 10⁻⁸ N in atoms to thousands of newtons in lightning. Before crunching numbers, estimate the order of magnitude. Two 1 μC charges at 1 m: F ≈ 9×10⁹ × 10⁻⁶ × 10⁻⁶ / 1 = 9×10⁻³ N — millinewton scale. If your calculator says 9000 N, you've misplaced a decimal. Order-of-magnitude estimates catch calculator entry errors instantly.

Apply the principle of superposition systematically. For N charges, the force on any one charge is the vector sum of forces from all (N-1) other charges. Work systematically: label charges 1, 2, 3... Calculate F₁₂, F₁₃, F₁₄... separately. Then add vectors. For symmetric arrangements, calculate one force and multiply by symmetry factors. For continuous charge distributions, integrate: F = ∫ k×q×dq/r².

Recognize when gravity is negligible.** For atomic and molecular scales, electric forces dominate gravity by factors of 10³⁰ or more. The electric force between a proton and electron is ~10⁻⁸ N; their gravitational attraction is ~10⁻⁴⁷ N — utterly negligible. Only for macroscopic neutral objects (where electric forces cancel) does gravity matter. In charged particle problems, ignore gravity unless specifically told otherwise.