Coulomb's Law & the Electric Field

Forces between point charges and the field that surrounds them

ElectromagneticsElectrical Engineering Year 2Free preview
⏱️ About 14 min

Stick a balloon to the wall after rubbing it on your hair and you have already demonstrated Coulomb's law — an inverse-square force between electric charges that was measured with a torsion balance in 1785 and still underpins every antenna, capacitor, and transistor you will study.

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The big idea: Every electric charge exerts a force on every other charge, and this force follows a precise inverse-square law. By introducing the electric field — the force per unit charge a tiny test charge would feel — we shift from action-at-a-distance to a field picture where each charge fills space with a measurable influence.
🎯 By the end, you'll be able to
  • Apply Coulomb's law to compute the magnitude and direction of the electrostatic force between two point charges
  • Determine the net force on a charge using the superposition principle with vector addition
  • Define the electric field E and compute it for a point charge using E = kQ/r²
  • Explain why the electric field is a property of the source charge, not the test charge

Coulomb's Law

In 1785 Charles-Augustin de Coulomb used a sensitive torsion balance to measure the force between two charged spheres. His finding, now called Coulomb's law, states that the electrostatic force between two point charges is directly proportional to the product of the charge magnitudes and inversely proportional to the square of the distance between them. The force acts along the straight line joining the two charges.

For two point charges Q₁ and Q₂ separated by a distance r, the magnitude of the force is:

\[ F = k\frac{|Q_1 Q_2|}{r^2} \]
Coulomb's law: magnitude of the electrostatic force between two point charges.

Direction and Sign

The constant k is Coulomb's constant, with value k = 8.99×10⁹ N·m²/C². The absolute value in the formula gives the magnitude; the direction is determined by the signs of the charges. When both charges have the same sign, the force is repulsive. When the charges have opposite signs, the force is attractive.

It is worth pausing to appreciate the structure of this law. The 1/r² dependence is the same mathematical form as Newton's law of gravitation, but the electrostatic force is enormously stronger. The gravitational attraction between two protons is roughly 10³⁶ times weaker than their electrostatic repulsion. The reason gravity dominates at cosmic scales is simply that matter is almost perfectly electrically neutral, while mass has no negative counterpart.

🔑 Vector Form of Coulomb's Law

The full vector form of Coulomb's law is F12 = k Q₁Q₂/r² r̂12, where r̂12 is the unit vector pointing from Q₁ to Q₂. The sign of Q₁Q₂ handles direction automatically: a positive product gives a force along r̂12 (repulsion), while a negative product reverses it (attraction).

The Superposition Principle

When more than two charges are present, the net force on any one charge is the vector sum of the individual forces exerted by each of the other charges. Superposition is a vector addition — you must resolve each force into components, sum the components separately, and then recombine.

Superposition also tells us that the presence of a third charge does not alter the force that the first charge exerts on the second. Each pair-wise interaction is independent. This linearity is the foundation for the integral approach we will use with continuous charge distributions in the next lesson.

📝 Worked example: A point charge Q₁ = +3 µC is located 0.5 m from a point charge Q₂ = −2 µC. Find the magnitude of the electrostatic force on each charge and state whether the force is attractive or repulsive.
  1. Identify the given values: Q₁ = +3×10⁻⁶ C, Q₂ = −2×10⁻⁶ C, r = 0.5 m.
  2. Use Coulomb's law for the magnitude: F = k|Q₁Q₂|/r².
  3. Compute the product of magnitudes: |Q₁Q₂| = (3×10⁻⁶)(2×10⁻⁶) = 6×10⁻¹² C².
  4. Multiply by k: k|Q₁Q₂| = (8.99×10⁹)(6×10⁻¹²) = 0.05394 N·m².
  5. Divide by r²: F = 0.05394 / (0.5)² = 0.05394 / 0.25 = 0.2158 N.
  6. Since the charges have opposite signs, the force is attractive.
✓ F = 0.2158 N, attractive

From Force to Field

Coulomb's law tells us the force between two charges, but it is often more useful to describe the influence of a single source charge on its surroundings. To do this, we introduce a small positive test charge q at the point of interest and measure the force F on it. The electric field E is defined as the force per unit charge:

\[ E = \frac{F}{q} \]
Definition of the electric field as force per unit test charge.

Electric Field of a Point Charge

Substituting Coulomb's law into the field definition gives the electric field of a point charge Q at a distance r:

\[ E = k\frac{Q}{r^2} \]
Electric field magnitude of a point charge Q at distance r.

Interpreting the Field

The field points radially outward from a positive charge and radially inward toward a negative charge. Notice that E depends only on the source charge Q and the distance r — the test charge q has been divided out. The electric field is a property of the source charge and the point in space, not of whatever probe we use to measure it.

This is a conceptual shift worth emphasizing. In the Coulomb picture, charges reach across empty space and grab each other. In the field picture, a charge creates a field in the space around it, and that field exerts a force on any other charge placed in it. The unit of electric field is newtons per coulomb (N/C), equivalent to volts per meter (V/m).

✨ Why the Test Charge Must Be Small

The test charge q must be small enough that its own field does not disturb the source charge distribution whose field we are measuring. In practice, this means taking the limit as q approaches zero: E = lim(q→0) F/q.

✏️ Practice: A point charge Q = 5 µC creates an electric field in the surrounding space. Find the electric field magnitude at a distance r = 2 m from the charge.
N/C
Solution
  1. Use the point-charge field formula: E = kQ/r².
  2. Substitute: E = (8.99×10⁹)(5×10⁻⁶) / (2)².
  3. Compute the numerator: (8.99×10⁹)(5×10⁻⁶) = 44950.
  4. Divide by r² = 4: E = 44950 / 4 = 11237.5 N/C, rounded to 11238 N/C.

Check your understanding

1. Two point charges have the same sign. The force between them is:
Like charges repel. The force is directed away from each charge, pushing them apart.
2. The electric field of a positive point charge points:
A positive test charge would be repelled by a positive source charge, so the field points radially outward, away from the charge.
3. If the distance from a point charge is doubled, the electric field magnitude becomes:
Since E = kQ/r², doubling r quadruples r², so E decreases by a factor of 4.
🔎 Related lessons
Field of Continuous Charge Distributions
✅ Key takeaways
  • Coulomb's law gives the electrostatic force between two point charges: F = k|Q₁Q₂|/r², where k = 8.99×10⁹ N·m²/C².
  • Like charges repel; opposite charges attract. The force always acts along the line joining the two charges.
  • The superposition principle states that the net force on a charge is the vector sum of the individual forces from all other charges.
  • The electric field E = F/q is the force per unit charge a small positive test charge would feel; for a point charge, E = kQ/r².
➡️ So far we have dealt only with point charges. Real-world charge, however, is spread over wires, plates, and volumes. In the next lesson we extend Coulomb's law to continuous charge distributions, treating extended objects as integrals of infinitesimal point charges — and we will discover that some of these integrals are far harder than they look.
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