Coulomb's Law — Apply Coulomb's law to compute forces between point charges - Physics 2

SECTION 1

Historical Context & Motivation

Long before physicists could probe the interior of the atom, natural philosophers puzzled over a deceptively simple question: what governs the force between electrically charged objects? Ancient Greeks such as Thales of Miletus noticed that rubbed amber attracted light objects, yet a rigorous quantitative law eluded investigators for over two millennia. The eventual answer—Coulomb's law—would become the electrostatic counterpart of Newton's gravitational law and serve as the cornerstone upon which all of classical electromagnetism rests. Understanding its historical development illuminates why the law takes the mathematical form it does and why it remains indispensable in modern physics and engineering.

1600 • Gilbert's De Magnete

William Gilbert systematically distinguishes electric attraction from magnetism and coins the term electricus from the Greek word for amber, laying the conceptual groundwork for the study of charge.

1733 • Two Kinds of Charge

Charles François de Cisternay du Fay demonstrates that electric charge comes in two varieties—vitreous (positive) and resinous (negative)—and that like charges repel while unlike charges attract.

1785 • Coulomb's Torsion-Balance Experiments

Charles-Augustin de Coulomb publishes precise measurements using his torsion balance, confirming the inverse-square dependence of electrostatic force on distance and its proportionality to the product of the charges.

1832 • Gauss Generalizes the Result

Carl Friedrich Gauss reformulates the inverse-square law using the concept of electric flux, leading to Gauss's law—an integral form equivalent to Coulomb's law that proves more versatile for symmetric charge distributions.

1865 • Maxwell's Synthesis

James Clerk Maxwell unifies electricity, magnetism, and optics in his field equations. Coulomb's law emerges as the electrostatic limit of this broader framework, securing its place as a foundational result in electromagnetic theory.

Coulomb's 1785 experiments resolved the central quantitative question: how does the electrostatic force depend on charge magnitude and separation distance? His answer—a force proportional to the product of the charges and inversely proportional to the square of the distance—provided the first predictive, testable law of electrostatics and opened the door to the entire mathematical edifice of electromagnetic theory.

SECTION 2

Core Principles & Definitions

Before applying Coulomb's law quantitatively, it is essential to internalize the physical assumptions and definitions that underpin it. The law applies strictly to point charges—objects whose spatial extent is negligible compared to the distance separating them—or to spherically symmetric charge distributions (by the shell theorem, they behave as point charges externally). The force is a central force, meaning it acts along the line joining the two charges, and it obeys Newton's third law: the force on charge 1 due to charge 2 is equal in magnitude and opposite in direction to the force on charge 2 due to charge 1.

Point Charge Idealization

A charged body whose physical size is negligible compared to all relevant distances. This lets us treat the charge as located at a single point, greatly simplifying force calculations.

Inverse-Square Dependence

The electrostatic force decreases as the square of the separation distance. Doubling the distance reduces the force to one-quarter of its original value, a signature of three-dimensional isotropic spreading.

Superposition Principle

When multiple charges are present, the net force on any one charge is the vector sum of the individual pairwise Coulomb forces. There is no shielding or saturation; each pair contributes independently.

Sign Convention & Force Direction

Like-sign charges (both positive or both negative) repel; opposite-sign charges attract. The algebraic sign of the product q₁q₂ determines the nature of the force—positive for repulsion, negative for attraction.

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KEY TAKEAWAY

Think of Coulomb's law as the electrical analog of Newton's law of gravitation. Just as two masses attract each other with a force proportional to the product of their masses divided by the square of their separation, two charges exert a force proportional to the product of their charges divided by the square of their separation. The crucial difference is that gravity is always attractive, whereas the electric force can attract or repel depending on the signs of the charges—making electrostatics far richer in its dynamical possibilities.

SECTION 3

Visual Explanation — Force Between Two Charges

Upper panel: two positive charges repel, with force vectors F₁₂ and F₂₁ directed away from each other along the line of separation r. Lower panel: a positive and a negative charge attract, with force vectors directed toward each other. In both cases |F₁₂| = |F₂₁| by Newton's third law.

The diagram above captures the two fundamental scenarios of Coulomb's law. In the upper panel, two like charges experience a repulsive force that pushes them apart; the force vectors point away from the opposing charge. In the lower panel, opposite charges experience an attractive force that draws them together. Crucially, both forces act along the line connecting the charge centers, and their magnitudes are equal—a direct manifestation of Newton's third law. The separation r is measured center-to-center, and the inverse-square scaling means even modest changes in r produce dramatic changes in force magnitude.

SECTION 4

Mathematical Framework

Coulomb's law can be expressed in both scalar (magnitude-only) and vector forms. The scalar form is sufficient for computing the strength of the interaction, while the vector form is indispensable for problems involving the superposition of forces from multiple charges. We begin with the scalar form and then generalize.

COULOMB'S LAW — SCALAR FORM

F = k × |q₁| × |q₂| / r²

F = magnitude of electrostatic force (N); k = Coulomb constant ≈ 8.99 × 10⁹ N·m²/C²; q₁, q₂ = point charges (C); r = center-to-center separation (m). Absolute values ensure we compute a positive magnitude; the direction is determined by sign analysis separately.

COULOMB'S LAW — VECTOR FORM

F⃗₁₂ = k × q₁ × q₂ / r² × r̂₁₂

F⃗₁₂ = force on charge 1 due to charge 2; r̂₁₂ = unit vector pointing from charge 2 toward charge 1. Note that q₁ and q₂ carry their signs here: a positive product yields repulsion (force along r̂₁₂), a negative product yields attraction (force opposite to r̂₁₂).

COULOMB CONSTANT — ALTERNATIVE FORM

k = 1 / (4πε₀) where ε₀ ≈ 8.854 × 10⁻¹² C²/(N·m²)

ε₀ is the permittivity of free space. This form connects Coulomb's law to Gauss's law and Maxwell's equations. In a dielectric medium with relative permittivity ε_r, replace ε₀ with ε₀ε_r to account for the medium's screening effect.

SUPERPOSITION PRINCIPLE

F⃗_net = Σᵢ F⃗ᵢ = Σᵢ k × q × qᵢ / rᵢ² × r̂ᵢ

The net force on a charge q in the presence of N other charges is the vector sum of each pairwise Coulomb force. Each term is computed independently, then the vectors are added component-by-component.

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Unit Reminder

Charge must be in coulombs (C) and distance in meters (m) when using k = 8.99 × 10⁹ N·m²/C². A common source of error is forgetting to convert microcoulombs (1 μC = 10⁻⁶ C) or nanocoulombs (1 nC = 10⁻⁹ C) before substituting into the formula. Similarly, distances given in centimeters must be converted to meters.

SECTION 5

Superposition & Multi-Charge Systems

Most physically interesting situations involve more than two charges, which necessitates the principle of superposition. Superposition states that the total force on a given charge is obtained by computing each pairwise Coulomb interaction independently and then performing a vector addition of all contributions. This seemingly simple rule is remarkably powerful: it allows us to handle arbitrarily complex charge configurations by reducing them to a sum of two-body problems. The critical procedural step is to resolve each force vector into Cartesian components, sum the x-components and y-components separately, and then recombine using the Pythagorean theorem and inverse tangent to find the resultant magnitude and direction.

Three charges arranged along perpendicular axes. Charge q₁ (cyan) experiences an attractive force F₁₂ toward q₂ (pink, along +x) and a repulsive force F₁₃ away from q₃ (amber, along −y). The dashed cyan arrow shows the net force obtained by vector addition of these two perpendicular components.

In the diagram above, note how the two individual force vectors are perpendicular, which makes the vector addition particularly clean—the net force magnitude is simply √(F₁₂² + F₁₃²). In general, however, the forces will not be perpendicular, and you must resolve each force into x- and y-components using trigonometry before summing. The procedure is: (1) identify all charges and their positions, (2) compute each pairwise force magnitude using the scalar form of Coulomb's law, (3) determine the direction of each force (attractive toward, repulsive away from the source charge), (4) decompose into components, (5) sum components, and (6) compute the resultant magnitude and angle.

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Common Pitfall

When resolving forces into components, students frequently confuse the angle at which the force acts. Always draw the geometry explicitly and measure angles from a consistent reference axis (usually the positive x-axis). The direction of each Coulomb force is along the line connecting the two charges involved—not along any coordinate axis unless the charges happen to be aligned with one.

SECTION 6

Worked Example — Three Collinear Charges

Consider three point charges arranged along the x-axis: q₁ = +6.0 μC at x = 0, q₂ = −3.0 μC at x = 0.20 m, and q₃ = +4.0 μC at x = 0.50 m. Find the net electrostatic force on q₂.

Net Force on q₂ from q₁ and q₃

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Step 1 — Identify Given Values and Convert Units

q₁ = +6.0 μC = +6.0 × 10⁻⁶ C at x = 0 m. q₂ = −3.0 μC = −3.0 × 10⁻⁶ C at x = 0.20 m. q₃ = +4.0 μC = +4.0 × 10⁻⁶ C at x = 0.50 m. The separation between q₁ and q₂ is r₁₂ = 0.20 m, and between q₂ and q₃ is r₂₃ = 0.50 − 0.20 = 0.30 m. The Coulomb constant is k = 8.99 × 10⁹ N·m²/C².

r₁₂ = 0.20 m; r₂₃ = 0.30 m
2
Step 2 — Compute |F₂₁| (Force on q₂ due to q₁)

|F₂₁| = k × |q₂| × |q₁| / r₁₂² = (8.99 × 10⁹) × (3.0 × 10⁻⁶) × (6.0 × 10⁻⁶) / (0.20)². The numerator is 8.99 × 10⁹ × 18.0 × 10⁻¹² = 0.16182 N. The denominator is 0.04. Therefore |F₂₁| = 0.16182 / 0.04 = 4.05 N.

|F₂₁| ≈ 4.05 N
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Step 3 — Determine Direction of F₂₁

q₂ is negative and q₁ is positive, so the force on q₂ due to q₁ is attractive—directed toward q₁, which lies in the −x direction. Therefore F₂₁ points in the −x direction: F₂₁,x = −4.05 N.

F₂₁ directed toward −x
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Step 4 — Compute |F₂₃| (Force on q₂ due to q₃)

|F₂₃| = k × |q₂| × |q₃| / r₂₃² = (8.99 × 10⁹) × (3.0 × 10⁻⁶) × (4.0 × 10⁻⁶) / (0.30)². The numerator is 8.99 × 10⁹ × 12.0 × 10⁻¹² = 0.10788 N. The denominator is 0.09. Therefore |F₂₃| = 0.10788 / 0.09 ≈ 1.20 N.

|F₂₃| ≈ 1.20 N
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Step 5 — Determine Direction of F₂₃

q₂ is negative and q₃ is positive, so the force is attractive—directed toward q₃, which lies in the +x direction. Therefore F₂₃,x = +1.20 N.

F₂₃ directed toward +x
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Step 6 — Sum Forces (Superposition)

Since all charges lie on the x-axis, both forces are along x. The net force on q₂ is F_net,x = F₂₁,x + F₂₃,x = −4.05 + 1.20 = −2.85 N. The negative sign indicates the net force is in the −x direction—q₂ is pulled toward q₁ more strongly than toward q₃.

F_net ≈ 2.85 N in the −x direction

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Sanity Check

The result makes physical sense: q₁ is closer to q₂ (0.20 m vs. 0.30 m) and carries a larger charge magnitude (6.0 μC vs. 4.0 μC), so the attraction toward q₁ dominates. Always verify that the direction and relative magnitude of your answer align with qualitative expectations before reporting.

SECTION 7

Strengths, Limitations & Common Misconceptions

Coulomb's law is among the most precisely verified laws in all of physics—experimental tests confirm the exponent of 2 in the inverse-square law to within ±10⁻¹⁶. Nonetheless, every physical law has a domain of validity, and recognizing the boundaries of Coulomb's law prevents misapplication. The following table summarizes its key strengths alongside important caveats.

Strengths	Limitations / Caveats
Exact for static point charges in vacuum—one of the most precisely confirmed laws in physics.	Applies rigorously only to static (or slowly moving) charges; for charges in rapid motion, the full Lorentz force and retarded potentials must be used.
Superposition makes it extensible to arbitrary charge distributions via integration (continuous) or summation (discrete).	Extended charge distributions require integration over the charge density, which can be analytically intractable for low-symmetry geometries.
Directly analogous to Newton's gravitational law, enabling transfer of intuition between the two domains.	Unlike gravity, the electric force can be screened by conductors or weakened in dielectric media, so bare Coulomb's law must be modified (ε₀ → ε₀εᵣ) in such environments.
Forms the basis for deriving Gauss's law, electric potential, and the field concept—serving as the gateway to all of electrostatics.	At subatomic scales (≲ 10⁻¹⁵ m), quantum electrodynamics (QED) replaces the classical law; virtual particle exchange modifies the effective force at very short distances.

Strengths and limitations of Coulomb's law

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COMMON MISCONCEPTION

A frequent error is assuming that Coulomb's law gives the net force when more than two charges are present. In reality, Coulomb's law describes only the pairwise interaction between two charges. For multi-charge systems, you must apply the law to every pair separately and then invoke superposition to obtain the net force. Another pitfall is neglecting to treat force as a vector—adding magnitudes without accounting for direction can yield dramatically incorrect results, especially when forces partially cancel.

SECTION 8

Connection to Electric Fields & Advanced Theory

While Coulomb's law gives a direct charge-to-charge force description, modern electrostatics reframes the interaction using the concept of the electric field. Rather than saying charge q₁ exerts a force on charge q₂ across empty space, we say q₁ creates a field E⃗ that permeates all of space, and q₂ then interacts locally with that field via F⃗ = q₂E⃗. This field perspective, while equivalent for electrostatics, becomes essential once charges move and electromagnetic waves propagate, because it naturally incorporates the finite speed of information transfer (the speed of light). Coulomb's law can therefore be viewed as the static, instantaneous limit of the full field-theoretic description encoded in Maxwell's equations.

Feature	Coulomb's Law (Force)	Electric Field Approach
Fundamental equation	F⃗ = kq₁q₂/r² r̂	E⃗ = kq/r² r̂; F⃗ = qE⃗
Requires knowledge of	Both charges and their separation	Source charge(s) to define E⃗; test charge separately
Best suited for	Small number of discrete point charges	Continuous distributions, superposition integrals, and time-varying fields
Generalizes to	Gauss's law (integral form)	Maxwell's equations, electromagnetic waves, QED
Action paradigm	Action at a distance (instantaneous)	Local field interaction (finite propagation speed)

Coulomb's law vs. the electric field formulation

As you proceed through Physics 2, you will encounter Gauss's law, which provides an equivalent but often more computationally elegant route to finding the electric field for highly symmetric charge distributions (spheres, infinite planes, infinite cylinders). Later, electric potential will offer a scalar alternative to the vector field, simplifying energy calculations. All of these formulations trace their roots back to Coulomb's inverse-square law; mastering it now ensures a solid foundation for every topic that follows.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Two identical positive point charges are separated by a distance d. A third positive charge is placed exactly at the midpoint. Describe the net electrostatic force on the middle charge and explain your reasoning using symmetry and the superposition principle.

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By symmetry, the middle charge is equidistant from both outer charges, and both outer charges carry the same sign and magnitude. Each exerts a repulsive force on the middle charge of equal magnitude, but these forces point in exactly opposite directions along the line connecting the three charges. By superposition, the vector sum of two equal-magnitude, antiparallel forces is zero. Therefore the net electrostatic force on the middle charge is zero. Note, however, that this equilibrium is unstable: any lateral displacement would produce a net force pushing the middle charge further from the axis.

Question 2: BASIC CALCULATION

Two point charges, q₁ = +5.0 μC and q₂ = −8.0 μC, are separated by 0.40 m in vacuum. Calculate the magnitude of the electrostatic force between them and state whether the force is attractive or repulsive.

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Using F = k|q₁||q₂|/r², substitute: F = (8.99 × 10⁹)(5.0 × 10⁻⁶)(8.0 × 10⁻⁶)/(0.40)². The numerator is 8.99 × 10⁹ × 40.0 × 10⁻¹² = 0.3596 N. The denominator is 0.16 m². Therefore F = 0.3596/0.16 ≈ 2.25 N. Since one charge is positive and the other is negative, the force is attractive—each charge is pulled toward the other.

Question 3: INTERMEDIATE

Three charges are arranged in a right triangle: q₁ = +2.0 μC at the origin, q₂ = +3.0 μC at (0.30, 0) m, and q₃ = −4.0 μC at (0, 0.40) m. Find the magnitude and direction of the net force on q₁.

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First, compute |F₁₂| = k(2.0 × 10⁻⁶)(3.0 × 10⁻⁶)/(0.30)² = (8.99 × 10⁹)(6.0 × 10⁻¹²)/0.09 = 0.599 N. Both charges are positive, so this force is repulsive and directed in the −x direction (away from q₂): F₁₂,x = −0.599 N, F₁₂,y = 0. Next, |F₁₃| = k(2.0 × 10⁻⁶)(4.0 × 10⁻⁶)/(0.40)² = (8.99 × 10⁹)(8.0 × 10⁻¹²)/0.16 = 0.449 N. q₁ is positive and q₃ is negative, so the force is attractive, directed toward q₃ in the +y direction: F₁₃,x = 0, F₁₃,y = +0.449 N. The net components are

F_x

= −0.599 N,

F_y

= +0.449 N. The magnitude is |F_net| = √(0.599² + 0.449²) = √(0.3588 + 0.2016) = √0.5604 ≈ 0.749 N. The direction is θ = arctan(0.449/0.599) ≈ 36.8° above the −x axis, or equivalently 180° − 36.8° = 143.2° measured from the +x axis.

Question 4: APPLIED

In a simplified model of a sodium chloride crystal, a Na⁺ ion (charge +e) and a Cl⁻ ion (charge −e) are separated by 0.282 nm. Calculate the magnitude of the Coulomb force between this ion pair, and compare it to the gravitational force between the ions (m_Na = 3.82 × 10⁻²⁶ kg, m_Cl = 5.89 × 10⁻²⁶ kg, G = 6.674 × 10⁻¹¹ N·m²/kg²).

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Electrostatic force:

F_E

= ke²/r² = (8.99 × 10⁹)(1.602 × 10⁻¹⁹)²/(2.82 × 10⁻¹⁰)² = (8.99 × 10⁹)(2.566 × 10⁻³⁸)/(7.952 × 10⁻²⁰) = 2.307 × 10⁻²⁸/7.952 × 10⁻²⁰ ≈ 2.90 × 10⁻⁹ N. Gravitational force:

F_G

= Gm₁m₂/r² = (6.674 × 10⁻¹¹)(3.82 × 10⁻²⁶)(5.89 × 10⁻²⁶)/(7.952 × 10⁻²⁰) ≈ 1.50 × 10⁻⁶¹/7.952 × 10⁻²⁰ ≈ 1.89 × 10⁻⁴² N. The ratio

F_E

F_G

≈ 1.53 × 10³³, demonstrating that the electrostatic force exceeds gravity by about 33 orders of magnitude at the atomic scale—this is why chemical bonding and crystal structure are governed entirely by electromagnetic forces, not gravity.

Question 5: CRITICAL THINKING

Two identical small conducting spheres carry charges Q and −3Q respectively and are fixed a distance d apart. They are briefly touched together and then returned to their original positions. By what factor does the magnitude of the electrostatic force between them change? Derive your answer algebraically and discuss the physical mechanism by which charge redistributes.

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Before contact, the force magnitude is

F_i

= k|Q||3Q|/d² = 3kQ²/d². When two identical conducting spheres touch, charge distributes equally between them: Q_total = Q + (−3Q) = −2Q, so each sphere acquires −2Q/2 = −Q. After separation, both carry charge −Q, and the force is

F_f

= k|−Q||−Q|/d² = kQ²/d². The ratio

F_f

F_i

= (kQ²/d²)/(3kQ²/d²) = 1/3. The magnitude decreases by a factor of 3. Additionally, the nature of the force changes: before contact the force was attractive (charges of opposite sign), but afterward both charges are negative, so the force becomes repulsive. The redistribution occurs because free electrons in conductors move under the influence of internal electric fields until equilibrium is reached—when the spheres touch, they form a single conductor, and charge flows until the potential is uniform across the combined surface.

SUMMARY

Lesson Summary

Summary

Coulomb's law quantifies the electrostatic force between two point charges: F = k|q₁||q₂|/r², where k ≈ 8.99 × 10⁹ N·m²/C² is the Coulomb constant and r is the center-to-center separation. The force is attractive for unlike charges and repulsive for like charges, and it obeys Newton's third law. The inverse-square dependence on distance means the force weakens rapidly with separation, dropping to one-quarter when the distance doubles.

For multi-charge systems, the superposition principle dictates that the net force on any charge is the vector sum of all pairwise Coulomb forces—each computed independently and then added component by component. Coulomb's law serves as the foundation of electrostatics and connects forward to the electric field concept, Gauss's law, and ultimately Maxwell's equations. Mastering its application—careful unit conversion, sign analysis, and vector decomposition—is the essential first step in your study of electromagnetic theory.