Physics 1B: Electric Force, Fields & Coulomb's Law

Electric charge — the fundamental property

Electric charge is a fundamental property of matter, just like mass. Unlike mass, however, charge comes in two varieties — positive and negative — and one charge can cancel another. This second feature is why we do not usually feel the enormous electric forces between the particles that make up ordinary matter: the positive and negative charges are almost exactly balanced.

Three facts about charge that the CBE tests:

• Like charges repel; unlike charges attract. Two positive charges push each other apart; a positive and a negative charge pull toward each other. This is the qualitative core of every electric-force question.
• Charge is conserved. In any physical process, the total charge before and after remains the same. Charge can move from one object to another (electrons flowing from a rubbed balloon to your hair), but it does not appear or disappear.
• Charge is quantized. The smallest unit of charge is the elementary charge, e ≈ 1.60 × 10⁻¹⁹ C, carried by a single electron (negative) or proton (positive). Every observed charge is an integer multiple of this basic unit.

The SI unit of charge is the coulomb (C). One coulomb is enormous — it takes about 6.24 × 10¹⁸ electrons to add up to a single coulomb of negative charge. Ordinary lab charges are typically nanocoulombs (10⁻⁹ C) or microcoulombs (10⁻⁶ C).

Coulomb's law — a familiar shape

The force between two point charges is described by Coulomb's law:

F = k · |q₁ · q₂| / d²

Where:

• F is the magnitude of the force in newtons (N).
• q₁ and q₂ are the two charges in coulombs (C).
• d is the distance between them in meters (m).
• k is Coulomb's constant: k ≈ 8.99 × 10⁹ N·m²/C². Note that k is much larger than G (6.67 × 10⁻¹¹) — this is why electric forces at atomic scales are so much stronger than gravity.

Take a moment to notice how similar this looks to Newton's law of universal gravitation. Both involve a constant times a product of two "somethings" divided by distance squared. This is not a coincidence: the mathematics of forces that spread out from a point in three-dimensional space naturally produces a 1/r² dependence. The differences: gravity uses masses (always positive), electricity uses charges (positive or negative), and the constants have wildly different magnitudes.

When using Coulomb's law on the CBE, focus on magnitudes with the absolute-value form given above, and use the like/unlike rule separately to determine whether the force is attractive or repulsive. This avoids messy sign-tracking for the direction.

Electric fields — force per unit charge

Working with Coulomb's law directly gets tedious when many charges are present. Physicists solve this by introducing the concept of a field: a region of space around a charge where any other charge would experience a force. The electric field at a location tells you how much force a test charge would feel there, per unit of test-charge magnitude:

E = F / q

Units are newtons per coulomb (N/C). Given a field E at some location, a charge q placed there feels force F = qE. The field itself is a vector, pointing in the direction a positive test charge would be pushed.

For a single point charge Q at distance r, the field magnitude is:

E = k · |Q| / r²

Direction: away from Q if Q is positive; toward Q if Q is negative. These directions define the field lines physicists draw around charges — arrows that trace the path a positive test charge would follow.

Field lines never cross (a single test charge cannot be pushed in two directions at once), and they always leave positive charges and terminate on negative charges (or extend to infinity).

Conductors and insulators

Materials fall into two broad categories based on how charge moves through them:

• Conductors — materials in which electrons can move freely. Metals (copper, silver, aluminum) are excellent conductors because their outer electrons are only loosely bound to individual atoms. In a conductor, any excess charge quickly spreads out over the surface until the forces on the electrons balance.
• Insulators — materials in which electrons are tightly bound and cannot move freely. Rubber, glass, wood, plastic, dry air. Excess charge stays where you put it (like the negative charge you deposit on a balloon by rubbing it on your hair).

The CBE tests three consequences of this distinction:

• Charge on a conductor moves to the outer surface. Interior electric field in a static conductor is zero.
• Two conductors touching share their charge freely until equilibrium.
• An insulator can retain a localized charge for a long time — which is why static electricity from a wool sweater can persist for hours in dry weather.

Conservation of charge — three ways charges move

Charge is transferred between objects through three mechanisms the CBE expects you to recognize:

• Conduction — direct contact allows charges to flow between conductors. Touching a charged rod to a metal sphere transfers some of the rod's charge to the sphere.
• Induction — a charged object is brought near (but not touching) a conductor, and the charges in the conductor rearrange in response. Ground the conductor while the charged object is nearby, and net charge can be trapped when you remove the ground.
• Polarization — even in an insulator, atoms can distort so that positive and negative sides shift slightly, creating temporary dipoles. This is why a charged balloon can stick to a wall: the wall becomes polarized in response.

In all three cases, total charge is conserved. Whatever charge one object gains, another loses. Charges are never created or destroyed, only moved.

Where students lose points

• Getting the direction of attraction/repulsion wrong. Like charges repel; unlike attract. Simple, but easy to slip when charges are mixed with each other in a chain.
• Forgetting to square the distance. Just like gravity, Coulomb's law is inverse-square. Doubling distance divides force by 4, not by 2.
• Confusing k with G. k = 8.99 × 10⁹ N·m²/C². G = 6.67 × 10⁻¹¹ N·m²/kg². The constants are 20 orders of magnitude apart.
• Missing that field E is not force F. E is force per unit charge. To get force, multiply E by q.
• Assuming charge stays where you put it on a conductor. It doesn't — charge spreads over the surface until equilibrium.

Worked example — force between two charges

Two point charges, q₁ = +5.0 × 10⁻⁶ C and q₂ = −3.0 × 10⁻⁶ C, are 0.10 m apart. Find the magnitude and describe the direction of the force between them. Use k = 8.99 × 10⁹ N·m²/C².

Step 1 — Apply Coulomb's law with magnitudes.

F = k · |q₁||q₂| / d² = (8.99 × 10⁹) × (5.0 × 10⁻⁶)(3.0 × 10⁻⁶) / (0.10)²

Step 2 — Compute the numerator: (8.99 × 10⁹) × (15 × 10⁻¹²) = 134.85 × 10⁻³ = 0.13485 N.

Step 3 — Divide by d² = 0.01 m². F = 0.13485 / 0.01 = 13.5 N.

Step 4 — Direction: unlike charges (one +, one −) attract. Force pulls the two charges toward each other along the line between them.

Check yourself

1. State the qualitative rule about like and unlike charges in one sentence.
2. Write Coulomb's law from memory and identify what each symbol represents.
3. Compare the constants k and G. Which force is stronger at everyday scales?
4. Define electric field in words, and give the relationship between F, E, and q.
5. Explain the difference between a conductor and an insulator.
6. Two identical +2 μC charges are 0.05 m apart. Estimate the force between them (order of magnitude is enough).

Practice with CBE-style questions

Coulomb's law and electric fields form the foundation of every subsequent Physics Semester B topic. Work through the practice bank filtered by Electric Force, Fields & Charge — each question includes a step-by-step solution and identifies the common error each distractor represents.

Independent practice content aligned to Texas Essential Knowledge and Skills (TEKS) §112.39(c)(5)(C), (c)(5)(D), (c)(5)(E). Not affiliated with TTU K-12, UT High School, UT-Austin, the Texas Education Agency, or any Credit by Examination administrator. Texas CBE™ does not administer any exam or grant academic credit.