Coulomb’s Law Calculator

Calculate the electric force between two point charges.

Chart: Electric Force vs. Distance (holding charges constant)

What is Electric Force and Coulomb’s Law?

The electric force is a fundamental force of nature that exists between any two electrically charged objects. This force can be either attractive or repulsive. If the charges have opposite signs (one positive, one negative), they attract each other. If they have the same sign (both positive or both negative), they repel each other. This is the principle behind many phenomena, from static cling to the chemical bonds that hold molecules together. Knowing how to calculate electric force using Coulomb’s law is essential for anyone studying physics or engineering.

Coulomb’s Law, first published by French physicist Charles-Augustin de Coulomb in 1785, provides a mathematical description of this force. The law 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. This relationship is known as an inverse-square law, similar in form to Newton’s Law of Universal Gravitation.

The Coulomb’s Law Formula Explained

The formula to calculate the electrostatic force (F) is expressed as:

F = k * |q₁ * q₂| / r²

This formula allows us to determine the magnitude of the force. The direction (attractive or repulsive) is determined by the signs of the charges.

Description of variables in the Coulomb’s Law formula.

Variable	Meaning	Standard Unit (SI)	Typical Range
F	The Electric Force	Newtons (N)	Varies widely (from micro-Newtons to many Newtons)
k	Coulomb’s Constant	N·m²/C²	Approx. 8.99 x 10⁹ N·m²/C² (in a vacuum)
q₁	Magnitude of the first charge	Coulombs (C)	Typically 10⁻⁹ C to 10⁻³ C in lab settings
q₂	Magnitude of the second charge	Coulombs (C)	Typically 10⁻⁹ C to 10⁻³ C in lab settings
r	Distance between the charges	Meters (m)	From micrometers (10⁻⁶ m) to several meters

Practical Examples

Example 1: Attractive Force

Let’s calculate the electric force between a proton and an electron in a hydrogen atom. This is a classic example of using a electrostatic force calculator.

• Input q₁ (proton): +1.602 x 10⁻¹⁹ C
• Input q₂ (electron): -1.602 x 10⁻¹⁹ C
• Input r (distance): 5.3 x 10⁻¹¹ m
• Calculation: F = (8.99 x 10⁹) * |(1.602 x 10⁻¹⁹) * (-1.602 x 10⁻¹⁹)| / (5.3 x 10⁻¹¹)²
• Result: The force is approximately 8.2 x 10⁻⁸ N. Because the charges are opposite, the force is attractive, holding the atom together.

Example 2: Repulsive Force

Imagine two small metallic spheres, each carrying a positive charge of +5 µC (microcoulombs), held 30 cm apart. We can find the repulsive force between them.

• Input q₁: +5 x 10⁻⁶ C
• Input q₂: +5 x 10⁻⁶ C
• Input r: 0.30 m
• Calculation: F = (8.99 x 10⁹) * |(5 x 10⁻⁶) * (5 x 10⁻⁶)| / (0.30)²
• Result: The force is approximately 2.5 N. Since the charges are the same, the force is repulsive, pushing the spheres apart.

How to Use This Electric Force Calculator

Using this calculator is straightforward. Here is a step-by-step guide on how to calculate electric force using Coulomb’s law with this tool:

1. Enter Charge q₁: Input the value for the first charge. Use a negative number for a negative charge. Select the appropriate unit (Coulombs, µC, etc.) from the dropdown menu.
2. Enter Charge q₂: Input the value for the second charge and select its unit.
3. Enter Distance r: Input the distance separating the centers of the two charges and select the distance unit (m, cm, mm).
4. View Results: The calculator will instantly update, showing the primary result for the electric force in Newtons (N). It will also state whether the force is attractive or repulsive and show the intermediate values used in the calculation.
5. Analyze the Chart: The chart dynamically illustrates how the force changes as distance changes, highlighting the inverse-square relationship.

Key Factors That Affect Electric Force

Several factors influence the strength of the electric force. Understanding them is key to mastering Coulomb’s Law. You might also be interested in a electric field calculator, which is closely related.

• Magnitude of Charges: The force is directly proportional to the product of the charges. Doubling one charge doubles the force. Doubling both charges quadruples the force.
• Distance Between Charges: This is the most critical factor. The force is inversely proportional to the square of the distance. Doubling the distance reduces the force to one-quarter (1/4) of its original value. Halving the distance increases the force by four times.
• Sign of Charges: The signs determine the direction of the force. Like charges (+ and + or – and -) result in a repulsive force, while opposite charges (+ and -) result in an attractive force.
• The Medium: Coulomb’s constant, k (≈ 8.99 x 10⁹ N·m²/C²), is the value for charges in a vacuum. If the charges are placed in a different medium (like oil or water), the force between them is reduced. This is described by the material’s permittivity.
• Presence of Other Charges: Coulomb’s law calculates the force between two point charges. If more charges are present, the net force on any one charge is the vector sum of the forces from all other charges (the principle of superposition).
• Shape and Size of Charged Objects: The formula is precise for “point charges” (charges whose physical size is negligible compared to the distance between them). For larger, extended objects, the calculation becomes more complex and often requires integration. However, for a uniformly charged sphere, it behaves as a point charge located at its center.

Frequently Asked Questions (FAQ)

1. What is Coulomb’s Constant (k)?

Coulomb’s constant (k) is a proportionality constant that relates the units of charge and distance to the unit of force. Its value in a vacuum is approximately 8.99 x 10⁹ N·m²/C². It is also expressed as 1 / (4πε₀), where ε₀ is the permittivity of free space.

2. What happens if one of the charges is zero?

If either q₁ or q₂ is zero, the product q₁*q₂ is zero, and therefore the electric force between them is zero. A charged object does not exert an electric force on an uncharged (neutral) object.

3. Why is the distance squared in the formula?

This is known as the inverse-square law. It arises because the influence of the electric field from a point charge spreads out uniformly in three-dimensional space. The surface area of a sphere is 4πr², so the field’s intensity (and thus the force it exerts) decreases in proportion to 1/r².

4. Can the electric force be negative?

When you calculate the force magnitude using |q₁ * q₂|, the result is always positive. A “negative” force in physics simply implies an attractive direction, which we determine from the signs of the charges. Our calculator explicitly states whether the force is “Attractive” or “Repulsive” to avoid confusion.

5. Is Coulomb’s Law always applicable?

It is highly accurate for static (non-moving) point charges. For moving charges, a more complete theory of electromagnetism, including magnetic fields, is required. It also works well for uniformly charged spheres when the distance is measured from their centers.

6. How does this differ from gravity?

Both are inverse-square laws, but the electric force is vastly stronger than gravity and can be either attractive or repulsive. Gravitational force is always attractive and depends on mass, not charge.

7. What is the smallest unit of charge?

The elementary charge (e), the charge of a single proton or electron, is the smallest unit of charge observed in nature. Its value is approximately 1.602 x 10⁻¹⁹ Coulombs. This is a fundamental concept often discussed alongside what is electric charge?.

8. What units should I use?

For the formula to work with the standard Coulomb’s constant, you must convert all units to SI units: charges in Coulombs (C), distance in meters (m), and the resulting force will be in Newtons (N). This calculator handles the unit conversions for you automatically.