Coulomb’s Law: The Formula That Measures the Force Between Charges 🧲

A more precise question 💭: We’ve said many times that charges attract or repel each other. But how much? What happens if we double the distance? What if we triple the charge? A Frenchman named Coulomb, using an incredibly sensitive balance, found the exact answer. Let’s get into it 👇

An Experiment with Hair-Strand Precision 🎯

Coulomb built a device called a torsion balance: a light rod suspended from its center by a thin fiber. When he brought a charged sphere near the sphere at the end of the rod, the rod rotated and the fiber twisted. By measuring the twist of the fiber, Coulomb could precisely measure very small forces. Clever, wasn’t it?

The result became this famous law:

Coulomb’s Law: The magnitude of the electric force between two point charges acts along the line joining them, is proportional to the product of their magnitudes, and inversely proportional to the square of the distance between them.

And in formula form:

$$ F = k,frac{|q_۱||q_۲|}{r^۲} tag{۱-۲} $$

where $q_1$ and $q_2$ are the charges (in Coulombs), $r$ is the distance (in meters), and $F$ is the force (in Newtons). And that $k$ is Coulomb’s constant:

$$ k approx ۹٫۰times۱۰^{۹} text{N·m}^۲/text{C}^۲ $$

🤓 A nice detail: sometimes $k$ is written as $k = dfrac{1}{4pivarepsilon_0}$, where $varepsilon_0$ (the permittivity of free space) equals $8.85times10^{-12} text{C}^2/text{N·m}^2$. For now, $k = 9times10^9$ works just fine for us.

And remember (according to Newton’s third law): the force that the first charge exerts on the second is equal in magnitude and opposite in direction to the force that the second exerts on the first. 🔁

Play with this formula yourself 🎮

Instead of just memorizing the formula, let’s feel it. In this simulation, change the charges and the distance, and see how the force arrow grows and shrinks. Specifically, try this: halve the distance — how many times does the force increase? (It works on mobile too):

A surprising example: Why doesn’t the atomic nucleus fly apart? 💥

Here, Coulomb’s law reveals something profound to us (Example 1-2 from the textbook):

a) In a hydrogen atom, the electron and proton are approximately $r = 5.3times10^{-11} text{m}$ apart. The force between them is:
$$ F = kfrac{e^۲}{r^۲} = (۹times۱۰^۹)frac{(۱٫۶times۱۰^{-۱۹})^۲}{(۵٫۳times۱۰^{-۱۱})^۲} approx ۸٫۲times۱۰^{-۸} text{N} $$

b) Now, in the helium nucleus, the two protons are only $r = 2.4times10^{-15} text{m}$ apart:
$$ F = kfrac{e^۲}{r^۲} = (۹times۱۰^۹)frac{(۱٫۶times۱۰^{-۱۹})^۲}{(۲٫۴times۱۰^{-۱۵})^۲} approx ۴۰ text{N} $$

Look at that! This force is repulsive (since both are positive) and is much larger than in part a. So, the nucleus should explode and fly apart right now! 🤯 But it doesn’t… which means there must be another force holding the nucleus together. We call this the nuclear force — and this is one of the ways physicists discovered that there are other fundamental forces in nature.

💡 Good to know: Masses also exert gravitational forces on each other. But for the electron and proton in hydrogen, the gravitational force is on the order of $10^{-47} text{N}$ — meaning it is about $10^{40}$ times smaller than the electric force! That’s why, in the world of particles, gravity is almost negligible; but on the scale of planets (where charges are neutral), gravity takes the lead.

When We Have Multiple Charges: The Principle of Superposition ➕

What if we have several charges instead of just two? Good news: electric force is additive. The net force on any charge is equal to the vector sum of the forces exerted by each of the other charges (as if the others weren’t there). This is known as the principle of superposition:

$$ vec{F}_text{net} = vec{F}_1 + vec{F}_2 + cdots + vec{F}_n $$

In this book, examples are always set up so that forces are either along a single line or perpendicular to each other — so the vector sum remains simple. 😌

🧮 Example 1-3 (from the book): Three charges $q_1=+2.5,mutext{C}$, $q_2=-1.0,mutext{C}$ and $q_3=+4.0,mutext{C}$ are on a straight line; the distance from A to B is 4 meters and from B to C is 2 meters. What is the net force on $q_3$?
– $F_{13} = (9times10^9)dfrac{(2.5times10^{-6})(4.0times10^{-6})}{(6)^2} = 2.5times10^{-3} text{N}$ (repulsive)
– $F_{23} = (9times10^9)dfrac{(1.0times10^{-6})(4.0times10^{-6})}{(2)^2} = 9.0times10^{-3} text{N}$ (attractive)
These two forces are in opposite directions, so: $F = 9.0times10^{-3} – 2.5times10^{-3} = 6.5times10^{-3} text{N}$, in the direction of the larger force (towards the left). Thus, $vec{F} = (-6.5times10^{-3} text{N}),hat{i}$.

🧮 Try it yourself (Exercise 1-2): In the same example above, find the net force on $q_1$.

Informal Summary 🎁

• Force between two point charges: $F = kfrac{|q_1||q_2|}{r^2}$, with $k = 9times10^9$.
• Force is directly proportional to the product of the charges and inversely proportional to the square of the distance. (Distance halved → Force quadrupled!)
• Force acts along the line connecting the two charges; like charges repel, unlike charges attract.
• Multiple charges? Use superposition (vector sum).

Fun Fact 💡 — Coulomb’s Name on the Eiffel Tower!

Charles-Augustin de Coulomb (1736–1806) published his law in 1785. He was so important to France that his name was engraved on the Eiffel Tower alongside 71 other scientists and engineers! Next time you see a picture of the Eiffel Tower, remember that the name of the person behind this formula is up there somewhere 🗼.

🔗 For the Curious — Deeper Study

• PhET — University of Colorado (Interactive): The same simulation as above with more activities to explore the relationship between force, charge, and distance → phet.colorado.edu
• HyperPhysics — Coulomb’s Law (Level: Intermediate, English): The formula, Coulomb’s constant, and comparison with gravity, complete with a calculator → hyperphysics.phy-astr.gsu.edu

Test Yourself 📝

Click on each question to reveal the answer 👇

In the next section (1-4), we dive into a revolutionary idea: instead of “force at a distance,” we describe the space around a charge using the concept of the electric field. This is where physics gets really beautiful! 👋

q1 = float(PARAMS.get(“q1”, 2.0)) * 1e-9
q2 = float(PARAMS.get(“q2”, 3.0)) * 1e-9
r_cm = float(PARAMS.get(“r_cm”, 10.0))
r = r_cm * 1e-2
k_e = 8.99e9

F = k_e * abs(q1 * q2) / r**2
F_type = ‘Repulsive’ if q1*q2 > 0 else ‘Attractive’

r_arr = np.linspace(0.5e-2, 0.6, 400)
F_arr = k_e * abs(q1 * q2) / r_arr**2

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4.5))

# Force vs distance
ax1.plot(r_arr*100, F_arr*1000, ‘#0073aa’, lw=2.2)
ax1.axvline(r_cm, color=’#e74c3c’, lw=1.8, ls=’–‘, label=f’r={r_cm}cm F={F*1000:.3f}mN’)
ax1.scatter([r_cm], [F*1000], color=’#e74c3c’, s=80, zorder=5)
ax1.set_xlabel(‘Distance r (cm) | Distance (cm)’)
ax1.set_ylabel(‘Force F (mN) | Force (mN)’)
ax1.set_title(f’Coulomb’s Law: F = kq₁q₂/r²n{F_type}’)
ax1.legend(fontsize=9); ax1.grid(alpha=0.3)
ax1.set_xlim(0, 60); ax1.set_ylim(0)

# Schematic
ax2.axis(‘off’); ax2.set_xlim(-1, 5); ax2.set_ylim(-1, 2)
c1 = ‘#e74c3c’ if q1 > 0 else ‘#3498db’
c2 = ‘#e74c3c’ if q2 > 0 else ‘#3498db’
ax2.add_patch(plt.Circle((0.5, 0.5), 0.35, color=c1))
ax2.add_patch(plt.Circle((3.5, 0.5), 0.35, color=c2))
ax2.text(0.5, 0.5, (‘+’ if q1>0 else ‘−’), ha=’center’, va=’center’, fontsize=20, color=’white’, fontweight=’bold’)
ax2.text(3.5, 0.5, (‘+’ if q2>0 else ‘−’), ha=’center’, va=’center’, fontsize=20, color=’white’, fontweight=’bold’)
ax2.text(0.