To find the electrostatic force between two charged objects, you use Coulomb’s Law: F = k × (q₁ × q₂) / r². This single equation covers the vast majority of electrostatic force problems you’ll encounter. The formula tells you the force depends on three things: how much charge each object carries, how far apart they are, and a universal constant that sets the scale.
Coulomb’s Law: The Core Formula
The equation looks like this:
F = k × (q₁ × q₂) / r²
- F is the electrostatic force, measured in newtons (N).
- q₁ and q₂ are the electric charges on the two objects, measured in coulombs (C).
- r is the distance between the centers of the two charges, measured in meters (m).
- k is Coulomb’s constant: approximately 8.99 × 10⁹ N·m²/C².
That constant, k, comes from a more fundamental value called the vacuum permittivity (ε₀ = 8.854 × 10⁻¹² F/m), where k = 1 / (4πε₀). For most problems, you can simply plug in 8.99 × 10⁹ and move on. Some textbooks write the formula as F = (1 / 4πε₀) × (q₁ × q₂) / r², which is mathematically identical.
Step-by-Step Calculation
Suppose you have two small spheres. One carries a charge of +3 microcoulombs (3 × 10⁻⁶ C), the other carries +2 microcoulombs (2 × 10⁻⁶ C), and they’re 0.5 meters apart.
Step 1: Identify your values. q₁ = 3 × 10⁻⁶ C, q₂ = 2 × 10⁻⁶ C, r = 0.5 m.
Step 2: Plug into the formula. F = (8.99 × 10⁹) × (3 × 10⁻⁶ × 2 × 10⁻⁶) / (0.5)².
Step 3: Multiply the charges. 3 × 10⁻⁶ × 2 × 10⁻⁶ = 6 × 10⁻¹². Square the distance: (0.5)² = 0.25.
Step 4: Finish the arithmetic. F = (8.99 × 10⁹) × (6 × 10⁻¹²) / 0.25 = 0.054 / 0.25 ≈ 0.216 N. That’s roughly the weight of a couple of grapes, pushing those two spheres apart.
How Distance Changes the Force
The r² in the denominator is what makes this an “inverse square law.” If you double the distance between two charges, the force drops to one-quarter of its original value. Triple the distance, and the force falls to one-ninth. This is the same type of relationship that governs gravity, and it means electrostatic force weakens rapidly as objects move apart but becomes extremely strong at close range.
This is why static electricity shocks happen at the last moment before you touch a doorknob. The charges on your hand and the knob exert a relatively mild force when your hand is a few centimeters away, but that force spikes as the gap shrinks to fractions of a millimeter.
Attractive vs. Repulsive Force
The sign of the charges tells you the direction of the force. Like charges (both positive or both negative) repel each other. Opposite charges (one positive, one negative) attract. When you multiply q₁ and q₂ in the formula, a positive result means repulsion and a negative result means attraction.
In practice, many textbook problems ask only for the magnitude of the force. In that case, use the absolute values of both charges and ignore the sign. If the problem asks for direction, keep the signs in, and the final positive or negative answer tells you whether the force pushes the charges apart or pulls them together.
When the Charges Aren’t in a Vacuum
Coulomb’s Law in its standard form assumes the two charges sit in empty space (or air, which behaves almost identically). If the charges are separated by a different material, like water, glass, or oil, the force between them is weaker. The material’s molecules partially cancel out the electric field, an effect measured by the material’s dielectric constant (also called relative permittivity).
To account for this, divide the force by the dielectric constant of the material. Water, for instance, has a dielectric constant around 80, so two charges submerged in water feel roughly 1/80th the force they’d feel in air. The modified formula becomes F = k × (q₁ × q₂) / (κ × r²), where κ is the dielectric constant.
Handling More Than Two Charges
When three or more charges are present, you find the net force on any one charge by calculating the force from each of the other charges individually, then adding those forces together as vectors. This is called the superposition principle. Each pair of charges interacts independently, so you apply Coulomb’s Law once for every pair that includes the charge you’re interested in.
The key here is that forces are vectors, meaning direction matters. You can’t just add the magnitudes. Instead, break each force into horizontal and vertical components, sum all the horizontal components together, sum all the vertical components together, then use the Pythagorean theorem to find the total magnitude. The angle of the net force comes from the arctangent of those two summed components. This part trips up a lot of students, so draw a diagram and label the direction of each individual force before doing any math.
How Electrostatic Force Compares to Gravity
Both Coulomb’s Law and Newton’s Law of Gravitation follow the inverse square relationship, but electrostatic force is overwhelmingly stronger. Between two electrons, the electrostatic repulsion is about 4.16 × 10⁴² times stronger than the gravitational attraction. Even between the heavier protons, the ratio is still about 1.24 × 10³⁶. Gravity only dominates at the scale of planets and stars because most large objects are electrically neutral, so their charges cancel out, leaving gravity as the only force still in play.
Where This Shows Up in Real Life
Coulomb’s Law isn’t just a textbook exercise. Photocopiers and laser printers rely on it. In a copier, a drum is given a uniform positive charge, then light from the document neutralizes the charge wherever the page is white. A negatively charged powder called toner sticks to the remaining positive areas, transferring the image. A sheet of paper with an even stronger positive charge then pulls the toner off the drum, and heated rollers fuse it permanently. Laser printers use the same principle, replacing the light exposure step with a precisely controlled laser beam.
Electrostatic precipitators in power plants use the same force to clean exhaust gases. Smoke and dust particles are given a positive charge, then passed through a negatively charged grid that attracts and holds them. Industrial precipitators remove over 99% of particles from smokestack emissions. Smaller versions of this technology appear in home air purifiers marketed as “electrostatic” or “ionic” filters.