Coulomb's Law Calculator
Calculate the electrostatic force between two point charges using Coulomb's Law (F = k·|q₁·q₂|/r²). Solve for force, charge, or distance with sign-aware attraction vs. repulsion detection, unit conversions, and an interactive force vs. distance chart. Perfect for physics students, engineers, and anyone working with electrostatics.
What Is Coulomb's Law?
Coulomb's Law (also known as Coulomb's inverse-square law) is a fundamental principle of physics that quantifies the electrostatic force between two stationary, electrically charged particles. It states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.
The law is expressed by the formula:
F = k × |q₁ × q₂| / r²
where:
- F = electrostatic force (in newtons, N)
- k = Coulomb constant ≈ 8.9875517923 × 10⁹ N·m²/C²
- q₁, q₂ = magnitudes of the two charges (in coulombs, C)
- r = distance between the charges (in meters, m)
The force is attractive if the charges have opposite signs (one positive, one negative) and repulsive if the charges have the same sign (both positive or both negative). This calculator detects the interaction type based on the signs of the input charges.
How Does the Coulomb's Law Calculator Work?
This calculator supports three calculation modes:
- Two Charges (F = k|q₁q₂|/r²): Calculate the electrostatic force given two charges and a distance. You can also solve for charge 1, charge 2, or distance by selecting the appropriate option.
- Solve for Charge: Given force, one known charge, and distance, calculate the unknown charge magnitude: |q₂| = F × r² / (k × |q₁|).
- Solve for Distance: Given force and two charges, calculate the separation distance: r = √(k × |q₁ × q₂| / F).
Basic Force: F = k × |q₁ × q₂| / r²
Unknown Charge: |q| = F × r² / (k × |q_known|)
Unknown Distance: r = √(k × |q₁ × q₂| / F)
Why Use This Coulomb's Law Calculator?
- Three Modes: Calculate force, unknown charge, or unknown distance.
- Flexible Solving: In basic mode, solve for force, charge 1, or charge 2.
- Attraction vs. Repulsion: Automatically detects and displays whether the force is attractive or repulsive.
- Force vs Distance Chart: Visualize the inverse-square relationship between force and distance.
- Step-by-Step Solution: See the full calculation broken down into intermediate steps.
- Unit Conversions: Supports multiple charge and distance units, plus force unit conversions.
- Formula Display: See the formula used for each calculation.
- Free & Private: No registration, no data storage.
The Coulomb Constant (k)
The Coulomb constant, also called the electrostatic constant, is defined as:
k = 1 / (4πε₀)
where ε₀ is the vacuum permittivity (also called the permittivity of free space), with a value of approximately 8.8541878128 × 10⁻¹² F/m.
The resulting value is k ≈ 8.9875517923 × 10⁹ N·m²/C². For most textbook problems, this is rounded to 9.0 × 10⁹ N·m²/C².
Understanding the Inverse-Square Law
Coulomb's Law is an inverse-square law, meaning the force decreases with the square of the distance. If you double the distance between two charges, the force becomes one-quarter of its original value. If you triple the distance, the force becomes one-ninth. This is visually apparent in the Force vs Distance chart generated by this calculator.
This inverse-square relationship is mathematically identical to Newton's Law of Universal Gravitation, which also follows an inverse-square law. However, the electrostatic force is vastly stronger than gravity — approximately 10³⁶ times stronger for the same distance and equivalent quantities.
Common Applications of Coulomb's Law
- Atomic Physics: Calculating the force between electrons and the nucleus, or between ions in a crystal lattice.
- Chemistry: Understanding ionic bonding, molecular geometry, and intermolecular forces.
- Electrical Engineering: Designing capacitors, electrostatic precipitators, and particle accelerators.
- Materials Science: Studying the behavior of charged particles in semiconductors and plasmas.
- Education: Teaching electrostatics, electric fields, and the fundamentals of electromagnetism.
Note: Coulomb's Law applies strictly to point charges at rest. For extended charged objects or moving charges, more advanced methods (such as Gauss's Law or integration) are required.
❓ Coulomb's Law Calculator FAQ
What is Coulomb's Law?
Coulomb's Law states that the electrostatic force between two stationary point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them: F = k × |q₁ × q₂| / r².
What is the formula for Coulomb's Law?
The fundamental formula is F = k × |q₁ × q₂| / r², where F is force in newtons, k is the Coulomb constant (≈ 8.99 × 10⁹ N·m²/C²), q₁ and q₂ are charges in coulombs, and r is distance in meters.
What is the Coulomb constant k?
The Coulomb constant is k = 1/(4πε₀) ≈ 8.9875517923 × 10⁹ N·m²/C². It is often rounded to 9.0 × 10⁹ N·m²/C² for textbook calculations. The constant relates the electrostatic force to the charges and distance.
What is the unit of electrostatic force?
In the SI system, force is measured in newtons (N). One newton is the force required to accelerate a 1 kg mass at 1 m/s². Other common units include dyne (dyn) and pound-force (lbf).
What is the difference between attraction and repulsion?
Attraction occurs when the two charges have opposite signs (one positive, one negative). Repulsion occurs when the charges have the same sign (both positive or both negative). This calculator automatically detects and displays the interaction type.
Can Coulomb's Law have negative force?
In vector form, the sign of the force indicates direction. In this calculator, we use the magnitude formula |F| and separately determine the interaction type (attractive or repulsive) based on the signs of the charges.
What is the inverse-square law?
An inverse-square law means a quantity is inversely proportional to the square of the distance. For Coulomb's Law, doubling the distance reduces the force to one-quarter of its original value. Tripling the distance reduces it to one-ninth.
How do I calculate the force between two charges?
Use the formula F = k × |q₁ × q₂| / r². Multiply the two charge magnitudes, multiply by the Coulomb constant, then divide by the square of the distance. This calculator does this automatically.
How do I calculate an unknown charge?
Rearrange Coulomb's Law: |q₂| = F × r² / (k × |q₁|). Multiply the force by the square of the distance, then divide by the product of the Coulomb constant and the known charge magnitude.
How do I calculate the distance between two charges?
Rearrange Coulomb's Law: r = √(k × |q₁ × q₂| / F). Multiply the Coulomb constant by the product of the charge magnitudes, divide by the force, then take the square root.
What are the units of charge?
The SI unit of charge is the coulomb (C). Common submultiples include millicoulomb (mC, 10⁻³ C), microcoulomb (µC, 10⁻⁶ C), nanocoulomb (nC, 10⁻⁹ C), and picocoulomb (pC, 10⁻¹² C).
What is the difference between Coulomb's Law and Newton's Law of Gravitation?
Both are inverse-square laws, but Coulomb's Law describes electrostatic forces (which can be attractive or repulsive) between charges, while Newton's Law describes gravitational forces (always attractive) between masses. The electrostatic force is vastly stronger — about 10³⁶ times stronger for equivalent quantities.
Does Coulomb's Law apply to all charged objects?
Coulomb's Law applies strictly to point charges at rest. For extended charged objects (like charged spheres or plates), you must integrate over the charge distribution unless the object is far away compared to its size, in which case it can be treated as a point charge.
What is the force between two 1 C charges separated by 1 meter?
Using F = k × |q₁ × q₂| / r², with q₁ = q₂ = 1 C, r = 1 m, and k = 8.99 × 10⁹ N·m²/C², the force is approximately 8.99 × 10⁹ N — about 9 billion newtons, an enormous force. This illustrates why 1 coulomb is a very large charge.
What is the force between a proton and an electron in a hydrogen atom?
A proton has charge +e and an electron has charge −e, where e = 1.602 × 10⁻¹⁹ C. At the Bohr radius (r ≈ 5.29 × 10⁻¹¹ m), the attractive force is approximately 8.2 × 10⁻⁸ N. This is the electrostatic force that holds the hydrogen atom together.
Can I use this calculator for real-world engineering applications?
Yes. This calculator is suitable for physics homework, engineering calculations, and real-world applications such as electrostatic precipitator design, capacitor analysis, and particle physics. Always verify results with professional tools for critical applications.