Why are there absolute value bars in the formula?

The absolute value bars ensure that the force you calculate is a positive number, representing just the magnitude (size) of the force. You figure out the direction (attractive or repulsive) separately by looking at the positive and negative signs of the charges.

What happens if there are more than two charges?

If there are three or more charges, you use the principle of superposition. You calculate the force from Coulomb's law between your main charge and each of the other charges one by one, and then add those force vectors together.

Does Coulomb's law work for moving charges?

Strictly speaking, Coulomb's law is for electrostatics, meaning the charges are stationary. If the charges are moving, especially at high speeds, you also have to account for magnetic forces and relativistic effects.

What is Coulomb's law?

Coulomb's law is a fundamental rule in physics that calculates the electric force between two stationary, electrically charged particles. In short, it tells us exactly how hard two charges push or pull on one another.

Think of holding two strong magnets in your hands. When you bring them close, they snap together or push apart strongly. If you move them further apart, that invisible push or pull fades very quickly. Coulomb's law describes this exact same behavior for electric charges, giving us the mathematical equation for the strength of that force.

The Formula

The precise definition of Coulomb's law is written as an equation: $F = k \frac{|q_1 q_2|}{r^2}$ . In this formula, $F$ is the electric force in Newtons. The variables $q_1$ and $q_2$ represent the amount of charge on each particle, measured in Coulombs (C). The $r$ stands for the distance between the centers of the two charges, measured in meters.

Finally, $k$ is Coulomb's constant. It is a fixed number that makes the units work out, approximately equal to $8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$ .

Like Charges Repel, Opposites Attract

Coulomb's law only gives us the magnitude (the size) of the force. To find the direction, you just need to look at the signs of the charges.

If both charges are positive or both are negative, they will push away from each other (repulsion). If one is positive and the other is negative, they will pull toward each other (attraction). You always apply the force along the straight line that connects the two charges.

The Inverse Square Rule

Notice that the distance, $r$ , is squared and on the bottom of the fraction. This is called an inverse-square law.

Imagine a flashlight beam: as you step back, the light spreads out and gets dimmer very fast. Similarly, if you double the distance between two charges, the electric force doesn't just cut in half—it drops to one-fourth of its original strength. If you triple the distance, the force becomes one-ninth as strong.

Where Students Slip Up

The most common mistake when using Coulomb's law is forgetting to convert units. Charges are often given in microCoulombs ( $\mu\text{C}$ ), which must be converted to Coulombs by multiplying by $10^{-6}$ before plugging them into the formula.

Another frequent error is forgetting to square the distance $r$ in the denominator. Always double-check your calculator entries to make sure that square is applied!

Worked through

Calculate the magnitude of the electric force between two point charges, $q_1 = 3 \mu\text{C}$ and $q_2 = -5 \mu\text{C}$ , separated by a distance of $0.2 \text{ m}$ . Is the force attractive or repulsive? (Use $k = 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$ ).

First, we convert the charges to standard units (Coulombs): $q_1 = 3 \times 10^{-6} \text{ C}$ $q_2 = -5 \times 10^{-6} \text{ C}$

Next, we use Coulomb's law to find the magnitude of the force. We use the absolute value of the charges because we only want the size of the force right now: $F = k \frac{|q_1 q_2|}{r^2}$ $F = (8.99 \times 10^9) \frac{|(3 \times 10^{-6})(-5 \times 10^{-6})|}{(0.2)^2}$ $F = (8.99 \times 10^9) \frac{15 \times 10^{-12}}{0.04}$ $F = (8.99 \times 10^9) (3.75 \times 10^{-10})$ $F \approx 3.37 \text{ N}$

Because $q_1$ is positive and $q_2$ is negative, they are opposite charges. Therefore, the force of $3.37 \text{ N}$ is attractive.

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Where this comes from: OpenStax Physics, Chapter 18: Electric Charge and Electric Field · Khan Academy, Physics Library: Electric charge and Coulomb's law · University Physics with Modern Physics by Young and Freedman