Coulomb's Law Calculator, Electrostatic Force
Calculate electrostatic force, charge, or distance using Coulomb's Law F = kq₁q₂/r². Supports multiple units, dielectric media, and solve-for-any-variable mode.
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What Is the Coulomb's Law Calculator, Electrostatic Force?
Coulomb's Law describes the force between two stationary point charges. Published by Charles-Augustin de Coulomb in 1785, it is to electrostatics what Newton's Law of Universal Gravitation is to mechanics, a fundamental inverse-square law. The force is along the line joining the two charges, attractive for opposite signs and repulsive for like signs.
The force scales linearly with each charge magnitude and falls off as the square of the separation distance. Doubling r reduces F by a factor of 4; halving r quadruples it. This 1/r² dependence is not a coincidence, it arises because electric field lines spread over the surface of a sphere whose area grows as 4πr².
In a medium other than vacuum, the force is reduced by the relative permittivity εᵣ (also called the dielectric constant). Water, with εᵣ ≈ 80, reduces electrostatic forces by 80× compared to vacuum, which is why salts dissolve readily in water but not in non-polar solvents.
Formula
Coulomb's Law, Electrostatic Force
Force: F = k · |q₁| · |q₂| / r²
With medium: F = (k/εᵣ) · |q₁| · |q₂| / r²
Solve for q₁: |q₁| = F · r² / (k · |q₂|)
Solve for r: r = √(k · |q₁| · |q₂| / F)
Related Quantities
Coulomb constant: k = 1/(4πε₀) = 8.9876 × 10⁹ N·m²/C²
Electric field: E = k|q₁|/r² (field of q₁ at distance r)
Potential energy: U = k·q₁·q₂/r (signed; negative = bound)
| Symbol | Meaning | Value / Units |
|---|---|---|
| F | Electrostatic force between the charges | Newtons (N); positive = repulsive |
| k | Coulomb constant (vacuum) | 8.9876 × 10⁹ N·m²/C² |
| ε₀ | Permittivity of free space | 8.854 × 10⁻¹² F/m |
| εᵣ | Relative permittivity (dielectric const) | dimensionless; εᵣ = 1 in vacuum |
| q₁, q₂ | Electric charges (signed) | Coulombs (C); use μC, nC, pC for small values |
| r | Distance between the charge centres | Metres (m) |
| E | Electric field at distance r due to q₁ | Volts per metre (V/m = N/C) |
| U | Electric potential energy of the system | Joules (J); negative for opposite charges |
How to Use
- 1Select solve-for: Choose what you want to find: Force F, Charge q₁, Charge q₂, or Distance r.
- 2Choose units: Select charge units (μC, nC, pC…) and distance units (m, cm, mm) that match your problem.
- 3Select medium: Choose the surrounding medium (vacuum, water, glass…) or enter a custom εᵣ value.
- 4Set charge signs: Use the +/− selectors next to each charge to indicate polarity (positive or negative).
- 5Enter known values: Fill in all fields except the one you are solving for. Use scientific notation (e.g. 1.6e-19).
- 6Calculate: Press "Calculate" or hit Enter. Results show force, electric field, potential energy, and a diagram.
Example Calculation
Example 1, Force between two point charges
Two charges, q₁ = +5 μC and q₂ = +10 μC, are placed 20 cm apart in air. What is the electrostatic force?
- ›k = 8.9876 × 10⁹ N·m²/C² (vacuum / air, εᵣ = 1)
- ›q₁ = 5 × 10⁻⁶ C, q₂ = 10 × 10⁻⁶ C, r = 0.20 m
- ›F = (8.9876 × 10⁹) × (5 × 10⁻⁶) × (10 × 10⁻⁶) / (0.20)²
- ›F = (8.9876 × 10⁹) × 5 × 10⁻¹¹ / 0.04
- ›F ≈ 11.23 N, repulsive (both charges positive)
Example 2, Hydrogen atom (proton–electron force)
The electron in a hydrogen atom is approximately 0.0529 nm (one Bohr radius) from the proton. Both carry charge magnitude e = 1.602 × 10⁻¹⁹ C with opposite signs.
- ›q₁ = +1.602 × 10⁻¹⁹ C, q₂ = −1.602 × 10⁻¹⁹ C, r = 5.29 × 10⁻¹¹ m
- ›F = k × e² / r²
- ›F = 8.9876 × 10⁹ × (1.602 × 10⁻¹⁹)² / (5.29 × 10⁻¹¹)²
- ›F ≈ 8.24 × 10⁻⁸ N ≈ 82.4 nN, attractive
- ›This force keeps the electron bound to the nucleus (centripetal electrostatic attraction).
Example 3, Solving for distance
Two charges of 1 μC each must be separated far enough so the repulsive force does not exceed 1 N. Find the minimum separation.
- ›r = √(k · |q₁| · |q₂| / F)
- ›r = √(8.9876 × 10⁹ × 10⁻⁶ × 10⁻⁶ / 1)
- ›r = √(8.9876 × 10⁻³)
- ›r ≈ 0.0948 m ≈ 9.48 cm
- ›If the charges are closer than 9.48 cm, the repulsive force exceeds 1 N.
Understanding Coulomb's Law, Electrostatic Force
Coulomb's Constant and the Permittivity of Free Space
The Coulomb constant k = 8.9876 × 10⁹ N·m²/C² is not a fundamental constant in modern physics, it is derived from ε₀, the permittivity of free space: k = 1/(4πε₀). The 4π factor arises naturally from the geometry of how electric field lines spread in three dimensions. In SI units, ε₀ is defined exactly as 8.8541878 × 10⁻¹² F/m.
When charges are embedded in a material, the electric field polarizes the molecules, partially cancelling the original field. This effect is captured by εᵣ. The table below shows typical values:
| Material | εᵣ (approx.) | Effect on Coulomb force |
|---|---|---|
| Vacuum | 1.000 | Reference, no reduction |
| Air (1 atm) | 1.0006 | Negligible difference from vacuum |
| Mica | 5–8 | Force reduced ~6× |
| Glass | 4–10 | Force reduced ~5–10× |
| Ethanol | 24.3 | Force reduced ~24× |
| Water (25 °C) | 80.1 | Force reduced ~80× |
| Barium titanate | 1200+ | Used in ceramic capacitors |
Attractive vs. Repulsive: The Sign Convention
The magnitude of the force is always positive. The direction depends on the signs of the charges: two positive or two negative charges repel; opposite charges attract. When building circuits or designing particle traps, knowing the direction is as important as knowing the magnitude.
The potential energy U = k·q₁·q₂/r carries a sign that encodes this: U < 0 for opposite charges means work must be done to separate them (they are in a bound state), while U > 0 for like charges means they naturally push apart.
Applications in Physics and Engineering
- ›Atomic physics: The Bohr model of hydrogen uses Coulomb attraction to set the electron orbit radius and energy levels.
- ›Capacitors: The charge stored on capacitor plates creates a Coulomb field in the gap, storing energy U = ½CV².
- ›Electrostatic precipitators: Industrial air cleaners charge particles electrostatically, then attract them to oppositely charged plates.
- ›Mass spectrometers: Ions are deflected by electric fields derived from Coulomb forces to separate them by mass-to-charge ratio.
- ›Protein folding: Charged amino acid residues interact via Coulomb forces in ionic solutions, shaping protein structure.
Frequently Asked Questions
What is Coulomb's Law used for?
- ›Coulomb's Law calculates the electrostatic force between two point charges.
- ›It is used in physics to analyze charged particle interactions in atoms and molecules.
- ›Engineers use it in capacitor design, electrostatic precipitators, and particle accelerators.
- ›It also underpins the electric field concept, the basis of all classical electromagnetism.
What is the Coulomb constant k?
- ›k = 8.9876 × 10⁹ N·m²/C², often written as 9 × 10⁹ for quick estimates.
- ›It equals 1/(4πε₀), where ε₀ = 8.854 × 10⁻¹² F/m is the permittivity of free space.
- ›The constant arose because the SI unit of charge (the Coulomb) was defined independently of force.
- ›In Gaussian units, k = 1 by definition, a cleaner but less practical system for everyday engineering.
How does the medium affect the electrostatic force?
- ›Inserting a dielectric material reduces the force by a factor of εᵣ (relative permittivity).
- ›Water has εᵣ ≈ 80, so charges in water experience forces ~80× weaker than in vacuum.
- ›This is why ionic compounds dissolve in water: the reduced attraction lets ions separate.
- ›For practical calculations, multiply k by (1/εᵣ) to get the effective Coulomb constant.
How is Coulomb's Law similar to Newton's Law of Gravitation?
- ›Both are inverse-square laws: force ∝ 1/r².
- ›Both involve a product of two quantities: charges (q₁q₂) or masses (m₁m₂).
- ›Coulomb's force can be ~10³⁶ times stronger than gravity between electrons.
- ›Unlike gravity (always attractive), Coulomb's force can be either attractive or repulsive.
What is electric potential energy in this context?
- ›U = k·q₁·q₂/r, the work done to assemble the two-charge configuration from infinity.
- ›Opposite charges (U < 0): energy must be added to separate them, they are bound.
- ›Like charges (U > 0): energy is released as they repel, they are unbound.
- ›U approaches zero as r → ∞, which is the conventional reference point.
Does Coulomb's Law work at all scales?
- ›Coulomb's Law is accurate for point charges or spherically symmetric charge distributions.
- ›At atomic scales (< 1 fm), quantum electrodynamics (QED) gives more accurate results.
- ›For moving charges, the full Maxwell equations (including magnetic effects) must be used.
- ›For most physics coursework and engineering applications, classical Coulomb theory is sufficient.
What units should I use for charge?
- ›The SI unit is the Coulomb (C). 1 C = charge on ~6.24 × 10¹⁸ protons.
- ›Practical charges are typically in μC (10⁻⁶ C), nC (10⁻⁹ C), or pC (10⁻¹² C).
- ›An electron/proton carries e = 1.602 × 10⁻¹⁹ C, about 0.16 pC.
- ›Use scientific notation in this calculator (e.g., 1.6e-19) for atomic-scale charges.
Why does the force increase so rapidly at short distances?
- ›Because force ∝ 1/r², halving the distance quadruples the force.
- ›At 1 cm, two 1 μC charges repel with ~90 N, about the weight of a 9 kg mass.
- ›At 1 mm, the same charges produce ~9,000 N, nearly 1 tonne of force.
- ›This rapid scaling is why charge separation at small distances powers capacitors and lightning.