Electrical Engineering & Physics · Coulomb’s Law
Coulomb’s Law – Formula, Units, and Electric Force Examples
Coulomb’s Law calculates the electric force between two charged particles using \(F = k\frac{|q_1 q_2|}{r^2}\), making it a foundational equation for electrostatics, electric field analysis, charge interactions, and engineering physics.
What is Coulomb’s Law? Formula and Definition
Coulomb’s Law gives the magnitude of the electric force between two point charges. The force increases with larger charge magnitude, decreases with the square of the separation distance, and acts along the line connecting the two charges.
Main formula
Use Coulomb’s Law to calculate the electrostatic attraction or repulsion between two charges.
Quick reference
- \(F\): electric force magnitude between the charges
- \(k\): Coulomb’s constant, approximately \(8.99\times10^9\ \text{N·m}^2/\text{C}^2\)
- \(q_1, q_2\): charge magnitudes in coulombs
- \(r\): distance between charge centers
Key takeaway
Most readers want this first: if the charges double, the force doubles for each charge term, but if the distance doubles, the force drops by a factor of four because the denominator is squared.
Coulomb’s Law is the electrostatics equivalent of an inverse-square force law. It tells you how strongly two charges push or pull on each other before you move into more advanced field, voltage, or multi-charge system analysis.
For engineering students, this equation is especially important because it builds the bridge from basic charge interactions to electric field strength, potential difference, capacitance concepts, and charge distribution models.
Editorial note: this page focuses on the standard electrostatics form of Coulomb’s Law for point charges or charge distributions that can reasonably be approximated as point charges. It is most reliable when the geometry and medium assumptions are clear.
Variables and units in Coulomb’s Law
Coulomb’s Law is simple in form, but the units and sign interpretation matter. In most calculations, the magnitude equation is used first and the force direction is determined separately from charge sign.
What each symbol means
| Symbol | Meaning | Typical unit | What it represents |
|---|---|---|---|
| \(F\) | Electric force | N | The magnitude of attraction or repulsion between two charges |
| \(k\) | Coulomb’s constant | \(\text{N·m}^2/\text{C}^2\) | The proportionality constant in vacuum or air approximations |
| \(q_1\) | First charge | C | Magnitude of the first electric charge |
| \(q_2\) | Second charge | C | Magnitude of the second electric charge |
| \(r\) | Separation distance | m | Distance between the charge centers |
| \(\varepsilon_0\) | Permittivity of free space | F/m | Used to define Coulomb’s constant more fundamentally |
| \(\varepsilon_r\) | Relative permittivity | dimensionless | Accounts for how a dielectric medium reduces electrostatic force |
| \(\hat{r}\) | Unit vector | dimensionless | Gives the direction of the force along the line between the charges |
Unit notes and conversion warnings
- Charge should be entered in coulombs, not microcoulombs, unless converted first.
- Distance must be in meters when using the standard SI value of \(k\).
- The force is in newtons when SI units are used consistently.
- Like charges repel and opposite charges attract, even if the magnitude equation uses absolute values.
Conversion cheat sheet
These charge conversions are one of the most common sources of algebra mistakes in Coulomb’s Law problems.
Helpful check: a very small change in distance can have a large effect on the result because force depends on \(1/r^2\).
The Inverse Square Law: How Distance Affects Electric Force
Coulomb’s Law is built around two central ideas: charge magnitude controls how strong the interaction can be, and distance controls how fast that interaction weakens. The result is an inverse-square relationship that becomes very strong at short range and much weaker at larger separation.
Defining Coulomb’s constant more precisely
In engineering physics, Coulomb’s constant is often written in terms of the permittivity of free space. This form is useful because it connects Coulomb’s Law to broader electrostatics topics such as field flux, dielectric materials, and capacitance.
This is the more fundamental form behind the familiar numerical constant used in introductory calculations.
Force grows with charge magnitude
If either charge increases, the electric force increases in direct proportion. If both charges are doubled, the force becomes four times larger because the product \(q_1 q_2\) appears in the numerator.
Distance controls the force most strongly
The separation term is squared, so distance matters strongly. Doubling the distance reduces the force to one-fourth, and tripling the distance reduces it to one-ninth.
This inverse-square behavior is one of the most important ideas to remember because it dominates the sensitivity of the result.
Medium and dielectric effects
The standard form is most often presented for vacuum or air, but the force changes when charges are placed in a medium such as oil, water, or another dielectric. In those cases, the relative permittivity reduces the force compared with vacuum.
This matters in capacitor design, insulation analysis, and electrostatic behavior inside real materials rather than ideal vacuum conditions.
Vector form and direction
In higher-level engineering physics, Coulomb’s Law is treated as a vector equation. The force acts along the line joining the two charges, and the unit vector indicates direction.
This form becomes especially useful in 2D and 3D force systems, superposition problems, and field derivations.
Comparison with Newton’s law of gravitation
Students often understand Coulomb’s Law more quickly when they compare it with gravity. Both are inverse-square laws, but electric forces can be vastly stronger than gravitational forces at particle scale.
| Feature | Coulomb’s Law | Newton’s Gravitation |
|---|---|---|
| General form | \(F = k\frac{|q_1 q_2|}{r^2}\) | \(F = G\frac{m_1 m_2}{r^2}\) |
| Source quantity | Electric charge | Mass |
| Can it repel? | Yes | No, gravity is attractive |
| Distance behavior | Inverse square | Inverse square |
| Relative strength | Usually far stronger at particle scale | Much weaker at particle scale |
Connection to electric field
Coulomb’s Law is also the foundation for the electric field equation of a point charge. If one charge is treated as the source and the other as a test charge, then electric field is force per unit charge.
This is why Coulomb’s Law is usually learned before electric field diagrams and field superposition.
When the simple form works best
The equation is most direct for point charges or spherically symmetric charge distributions. If the geometry is extended, irregular, or spread over a surface or volume, more advanced electrostatics methods are often needed.
For many introductory and engineering physics problems, Coulomb’s Law is the first and most useful tool for translating charge and geometry into force.
Worked examples using Coulomb’s Law
These examples cover the most common search intent: direct force calculation, distance sensitivity, and interpretation of attraction versus repulsion.
Example 1: Force between two small charges
Scenario: Two charges of \(3.0\times10^{-6}\ \text{C}\) and \(2.0\times10^{-6}\ \text{C}\) are separated by \(0.20\ \text{m}\). Find the magnitude of the electric force.
Answer: the force magnitude is about \(1.35\ \text{N}\).
Engineering interpretation: even microcoulomb-scale charges can create noticeable forces at short range. This is why electrostatic effects matter in sensors, charged surfaces, and insulating materials.
Example 2: Effect of doubling the separation distance
Scenario: If the charges in Example 1 stay the same but the separation increases from \(0.20\ \text{m}\) to \(0.40\ \text{m}\), find the new force.
Doubling the distance reduces the force by a factor of 4.
Answer: the new force is about \(0.338\ \text{N}\).
Engineering interpretation: the distance term usually drives the sensitivity of the result more strongly than most users expect. This is why electrostatic force changes quickly with spacing.
Example 3: Identifying attraction versus repulsion
Scenario: Two charges have values \(q_1=+5\ \mu\text{C}\) and \(q_2=-2\ \mu\text{C}\). They are separated by \(0.10\ \text{m}\). Find the force magnitude and determine whether the force is attractive or repulsive.
Answer: the magnitude is about \(8.99\ \text{N}\), and the force is attractive because the charges have opposite signs.
Engineering interpretation: in many practical problems, the magnitude and direction are both required. The formula gives the magnitude, while charge sign determines whether the bodies move toward or away from each other.
Mistakes, limits, and engineering checks
Coulomb’s Law is straightforward once the variables are clear, but the most common mistakes usually come from unit conversion, geometry assumptions, or sign interpretation.
Always convert microcoulombs and millimeters first
Many problems give charge in \(\mu\text{C}\), nC, or pC and distance in mm or cm. The standard SI value of \(k\) assumes coulombs and meters, so conversion must happen before calculation.
Do not ignore the inverse-square effect
Distance errors become amplified because the denominator is squared. A small geometry mistake can create a large force error.
Use the point-charge assumption appropriately
If the objects are large compared with the spacing, or if the charge distribution is complex, the point-charge model may not be accurate enough. That is a signal to move toward a field-based or distributed-charge method.
Separate force magnitude from direction
A good workflow is to calculate the magnitude first using absolute value, then determine attraction or repulsion from the signs of the charges. This avoids sign confusion in the algebra.
Fast sanity checks
- If either charge becomes zero, the electrostatic force must become zero.
- If distance increases, the force must decrease.
- If both charge magnitudes increase, the force magnitude must increase.
- Opposite charges attract and like charges repel.
Frequently asked questions about Coulomb’s Law
What is Coulomb’s Law formula?
Coulomb’s Law is \(F = k\frac{|q_1 q_2|}{r^2}\), where electric force equals Coulomb’s constant times the product of the charge magnitudes divided by the square of the separation distance.
What are the units used in Coulomb’s Law?
In SI units, force is in newtons, charge is in coulombs, and distance is in meters. Coulomb’s constant is typically expressed in \(\text{N·m}^2/\text{C}^2\).
Do opposite charges attract or repel?
Opposite charges attract, while like charges repel. The equation gives the magnitude, and the charge signs determine the direction of interaction.
Why does the force decrease so quickly with distance?
Because Coulomb’s Law is an inverse-square law. The force is proportional to \(1/r^2\), so increasing distance reduces the force rapidly.