Coulomb's Law and Electric Force

Lesson 2 of 5: Static Electricity

In this lesson:

• The formula for electric force between two charges
• How to determine direction — attract or repel
• Why electric force dwarfs gravity at atomic scales
• How to find the net force from multiple charges

What You Will Learn in This Lesson

By the end, you will be able to:

1. State Coulomb's Law in words and formula
2. Calculate electric force magnitude between two charges
3. Determine force direction using charge signs
4. Compare electric force to the gravitational force
5. Apply superposition to find the net force

You Already Know This Formula Shape

Newton's Law of Gravity — from sec-7-2:

• Force ∝ product of both masses
• Force ∝ 1/r² — inverse-square law

Today's formula has the same shape — different constants, different quantities.

Coulomb's Law: Force Between Two Charges

• = electric force magnitude (N)
• N·m²/C² ≈ N·m²/C²
• , = charge magnitudes (C)
• = distance between charges (m)

The formula gives magnitude only — direction comes from the sign rule.

How Force Drops with Distance

Doubling reduces force by 4. Tripling reduces force by 9.

Direction of the Electric Force

• Same sign (+ and + or − and −): repulsive — charges push apart
• Opposite sign (+ and −): attractive — charges pull together
• Newton's Third Law: force on equals force on in magnitude

Procedure: magnitude first, then direction from sign rule.

Worked Example: Finding Force Magnitude

Given: , , m

Step 1: Convert to coulombs:
C, C

Step 2: Apply Coulomb's Law:

Step 3: Direction — opposite signs → attractive

Distance Doubles — What Happens to Force?

Two positive charges experience force at distance .

If the distance doubles (), what happens to the force?

• A) Force halves →
• B) Force becomes one-quarter →
• C) Force doubles →

is squared — what happens to ?

Inverse-Square Scaling: The Correct Answer

B) Force becomes one-quarter:

If :

Remember: doubling divides force by 4, not 2 — because is squared.

Moving from One Pair to Comparing Forces

You can now calculate the electric force between any two charges.

Coming up: How does this force compare to gravity — the other inverse-square law?

The comparison reveals something remarkable about atomic scale.

Coulomb's Law vs. Newton's Gravitation

Both are inverse-square laws — but by a factor of .

Why Electric Force Dominates at Atomic Scale

• N·m²/C² vs. N·m²/kg²
• For a proton-electron pair:
• Electric force is 39 orders of magnitude stronger at atomic scale
• Gravity is irrelevant in chemistry and atomic physics

Worked Example: Proton-Electron Force Ratio

Proton: kg, C
Electron: kg, C, m

Why Large Objects Are Electrically Neutral

• Matter contains roughly equal positive (protons) and negative (electrons) charges
• Charge cancellation: positive and negative cancel → near-zero net charge
• Large objects (planets, stars) appear electrically neutral — gravity dominates
• Gravity has no cancellation — all mass contributes positively

Why Doesn't Gravity Dominate at Atomic Scale?

Why does electric force dominate instead of gravity?

• A) Protons are too small for gravity to act on
• B) Gravity only acts between large objects
• C) Electric force is ~10³⁹ times stronger at this scale
• D) Gravity is repulsive at small distances

From One Pair to Multiple Charges

You can calculate the force between any single pair using Coulomb's Law.

What if a charge is surrounded by several other charges?

Each pair still obeys Coulomb's Law independently — superposition handles the rest.

Superposition Principle for Electric Forces

• Calculate each pairwise force independently using Coulomb's Law
• Assign direction based on attract/repel for each pair
• Add all forces as vectors (sign = direction in 1D)

This course focuses on 1D superposition — forces along one line.

Three-Charge Superposition: Finding Net Force

Find net force on B (−2 μC). Rightward = positive.

Step 1: Force on B from Charge A

, , m

A attracts B → force on B points left (negative direction)

Step 2: Force on B from Charge C

, , m

C attracts B → force on B points right (positive direction)

Step 3: Net Force on Charge B

Result: Net force = 0.55 N to the right

The pull from C (closer, to the right) wins over the pull from A (farther, to the left).

Guided Practice: Net Force on Middle Charge

Three charges: C at , C at m, C at m.

Find the net force on the middle charge.

Step 1: Direction of each force?
Step 2: Magnitudes from Coulomb's Law
Step 3: Assign signs and sum

Direction Check: Force Between Opposite Charges

Charge A (+). Charge B (−). Force from A acts on B in which direction?

• A) Away from A — repulsive
• B) Toward A — attractive
• C) No force acts between unlike charges
• D) Depends on charge size

Repulsion or attraction — which rule applies?

Direction Check: Answer and Explanation

B) Toward A — attractive

• Opposite signs → attract
• Force on B points in the direction of A
• Magnitude calculated from
• Repulsion ≠ no force — a repulsive force still has magnitude and direction

Key Takeaways: Formula and Direction

✓ — both charges required

✓ Same sign → repels; opposite sign → attracts

✓ Newton's 3rd Law: both charges feel equal force

is the field formula — not the force formula

Repulsion = force pointing away — not zero force

Key Takeaways: Scaling and Superposition

✓ Doubling divides by 4 — inverse-square law

✓ Electric force ~ times stronger than gravity at atomic scale

✓ Charge cancellation makes large objects electrically neutral

✓ Superposition: pairwise forces calculated independently, then added

Doubling → , not — is squared

Coming Up: The Electric Field

Lesson 3 of 5: Electric Field

• What is the electric influence of a single charge on the space around it?
• The electric field — Coulomb's Law extended to one source charge

Prepare: understand the difference between E = kq/r² and F = k|q₁q₂|/r²