3aElectric Charge and Coulomb's Law
Explore the fundamental property of electric charge, the inverse-square law of electrostatic force, and how charge is transferred between objects.
Coulomb's law is the foundation of all electrostatics and, by extension, all of chemistry and materials science. The forces that hold atoms together in molecules, that give solids their structure, and that drive every chemical reaction are fundamentally electrostatic. Understanding charge is the gateway to understanding electricity, magnetism, and light.
Why does rubbing a balloon on your hair make it stick to the wall — and what fundamental force is responsible?
Lesson Overview
Electric charge is a fundamental property of matter. There are two types: positive (protons) and negative (electrons). Like charges repel; unlike charges attract. The SI unit of charge is the coulomb (C). The elementary charge is e = 1.602×10⁻¹⁹ C. Charge is quantized (always a multiple of e) and conserved (total charge in an isolated system is constant). Coulomb's law gives the electrostatic force between two point charges: F = kq₁q₂/r², where k = 8.99×10⁹ N·m²/C². This inverse-square law is analogous to Newton's law of gravitation. Conductors allow charge to flow freely; insulators do not. Charging methods include friction, conduction, and induction.
Key Equations
Worked Examples
Two point charges, q₁ = +3.0 μC and q₂ = −2.0 μC, are separated by 0.30 m. Find the magnitude and direction of the electrostatic force on each charge.
How many electrons must be removed from a neutral object to give it a charge of +1.0 μC?
Three charges are placed on a line: q₁ = +2.0 μC at x = 0, q₂ = −3.0 μC at x = 0.40 m, q₃ = +1.0 μC at x = 0.80 m. Find the net force on q₂.
Two identical conducting spheres each carry charge +Q. They are touched together and then separated. What is the charge on each sphere?
At what distance from a +5.0 μC charge is the electrostatic force on a +2.0 μC charge equal to 0.50 N?
Guided Problems
Two protons are separated by 1.0×10⁻¹⁰ m (roughly the diameter of an atom). Find the electrostatic force between them. Compare it to the gravitational force (m_p = 1.67×10⁻²⁷ kg, G = 6.67×10⁻¹¹ N·m²/kg²).
Hint: Use F_E = ke²/r² and F_G = Gm_p²/r². The ratio F_E/F_G shows how much stronger the electric force is.
A charge of −4.0 μC is placed at the origin. Where on the x-axis should a +6.0 μC charge be placed so that the net force on a +1.0 μC charge at x = 0.30 m is zero?
Hint: For the net force to be zero, the forces from both charges on the +1.0 μC charge must be equal and opposite. Set up the equation and solve for the position of the +6.0 μC charge.
A neutral metal sphere is brought near (but not touching) a positively charged rod. Describe the charge distribution on the sphere. What happens when the rod is removed?
Hint: This is induction. The rod attracts electrons to the near side, leaving the far side positive. When the rod is removed, the charges redistribute uniformly — the sphere remains neutral.
The force between two charges is 0.80 N. If the distance between them is doubled, what is the new force?
Hint: Coulomb's law is an inverse-square law: F ∝ 1/r². If r doubles, F decreases by a factor of 4.
Explain why a charged comb can attract small pieces of paper even though the paper is electrically neutral.
Hint: The charged comb induces a charge separation in the paper (polarization). The near side of the paper is attracted more strongly than the far side is repelled, giving a net attractive force.
Key Vocabulary
Electric Charge
A fundamental property of matter that causes it to experience electromagnetic forces. Exists as positive or negative; measured in coulombs (C). Like charges repel; unlike charges attract.
Example: A proton has charge +e = +1.602×10⁻¹⁹ C; an electron has charge −e = −1.602×10⁻¹⁹ C.
Conservation of Charge
The total electric charge in an isolated system remains constant. Charge can be transferred between objects but cannot be created or destroyed.
Example: When you rub a glass rod with silk, the rod becomes positive and the silk becomes equally negative — total charge remains zero.
Coulomb's Law
The electrostatic force between two point charges is F = kq₁q₂/r², where k = 8.99×10⁹ N·m²/C². The force is along the line joining the charges, attractive for opposite charges and repulsive for like charges.
Example: Two charges of 1 C separated by 1 m experience a force of ~9×10⁹ N — enormously strong, which is why macroscopic objects are nearly always electrically neutral.
Conductor
A material in which electric charge (electrons) can move freely. Metals are good conductors because their outer electrons are loosely bound.
Example: Copper wire conducts electricity because its electrons move freely through the metal lattice.
Insulator
A material in which charge cannot move freely. Electrons are tightly bound to their atoms.
Example: Rubber, glass, and plastic are insulators — they can hold a static charge without it flowing away.
Charging by Induction
A method of charging a conductor without direct contact. A charged object brought near a conductor causes charge separation; if the conductor is grounded (or split) while the charged object is near, a net charge remains.
Example: Bringing a negative rod near a neutral sphere and then grounding the sphere leaves the sphere with a positive charge after the rod is removed.
Workbook Check — Interactive Quiz
Interactive Practice — 5 Questions
Coulomb's law states that the force between two point charges is:
Two charges of +2 μC and +2 μC are separated by 0.10 m. The force between them is:
The elementary charge e equals:
If the distance between two charges is tripled, the force between them:
Which method of charging leaves the charging object unchanged?
Independent Practice
Two charges q₁ = +4.0 μC and q₂ = −6.0 μC are 0.50 m apart. Find the magnitude and direction of the force on each charge.
How many electrons are in 1.0 C of charge? How does this compare to the number of atoms in a gram of hydrogen?
Three charges are at the corners of an equilateral triangle with side 0.30 m: q₁ = +1.0 μC, q₂ = +1.0 μC, q₃ = −1.0 μC. Find the net force on q₃.
Explain the difference between charging by conduction and charging by induction. In which method does the charging object lose charge?
★ Two identical small spheres each have mass 0.10 g and are suspended by strings of length 20 cm from the same point. When given equal charges q, they hang at 10° from vertical. Find the charge q on each sphere. (Hint: use equilibrium of forces — tension, gravity, and electrostatic repulsion.)
ChallengeCommon Mistakes
Using Coulomb's law with charges in microcoulombs without converting to coulombs
Always convert charge to coulombs (C) before substituting into F = kq₁q₂/r². 1 μC = 10⁻⁶ C, 1 nC = 10⁻⁹ C.
Forgetting that Coulomb's law gives magnitude — the direction must be determined separately
F = kq₁q₂/r² gives the magnitude. Direction: like charges repel (away from each other); unlike charges attract (toward each other).
Thinking charge can be created or destroyed
Charge is conserved. When objects are charged by friction, equal and opposite charges are created — the total charge remains zero.
Applying Coulomb's law to extended charge distributions as if they were point charges
Coulomb's law applies exactly only to point charges. For extended objects, it is a good approximation only when the separation is much larger than the object size.
Math Tips
Coulomb's law: F = kq₁q₂/r². k = 8.99×10⁹ ≈ 9×10⁹ N·m²/C². Always use SI units (C, m, N).
Inverse-square law: doubling r → F/4; tripling r → F/9; halving r → 4F. Memorize this pattern.
For multiple charges, use superposition: find the force from each charge separately, then add as vectors (paying attention to direction).
Elementary charge: e = 1.6×10⁻¹⁹ C. Number of elementary charges: n = q/e. Always an integer for real objects.