1.2 Coulomb's law

Coulomb's law describes the force between two charged particles. It's one of the most important relationships in electrostatics and sets the foundation for understanding electric fields, atomic structure, and how charges behave in real systems.

Electrostatic force

The electrostatic force governs how charged particles push or pull on each other. Everything from the structure of atoms to the operation of electronic devices traces back to this interaction. In this course, Coulomb's law is your starting point for understanding electric fields and electric potential.

Definition of Coulomb's law

Coulomb's law states that the electrostatic force between two point charges is proportional to the product of their charges and inversely proportional to the square of the distance between them. It applies to point charges (or objects small enough to treat as points) in a vacuum or air.

The mathematical expression is:

$F = k\frac{q_1 q_2}{r^2}$

Each variable matters:

• $k$ is the Coulomb constant, approximately $8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$
• $q_1$ and $q_2$ are the magnitudes of the two charges, measured in Coulombs (C)
• $r$ is the distance between the centers of the two charges, measured in meters (m)
• $F$ is the resulting force, measured in Newtons (N)

A common mistake: students forget that $r$ is the distance between the charges, not the position of one charge. Also, if you plug in the signs of the charges, a negative result means attraction and a positive result means repulsion.

Vector nature of force

The electrostatic force isn't just a number; it has direction too. The force always acts along the straight line connecting the two charges. If the charges attract, the force on each charge points toward the other. If they repel, the force points away.

This also follows Newton's Third Law: the force that charge 1 exerts on charge 2 is equal in magnitude and opposite in direction to the force charge 2 exerts on charge 1. When you're working with charges that aren't along a single axis, you'll need to break the force into x and y components using trigonometry.

Charge interactions

Like vs. unlike charges

The sign of each charge determines whether the force is attractive or repulsive:

• Like charges (both positive or both negative) repel each other
• Unlike charges (one positive, one negative) attract each other

The strength of the interaction depends on the magnitudes of both charges. Two charges of $\pm 3 \text{ C}$ interact more strongly than two charges of $\pm 1 \text{ C}$ at the same distance.

Attractive vs. repulsive forces

• Attractive forces pull charges closer together, decreasing the distance between them
• Repulsive forces push charges apart, increasing their separation

Both types obey the same inverse square law. Doubling the distance between any two charges reduces the force to one-quarter of its original value, regardless of whether the force is attractive or repulsive.

Superposition principle

When more than two charges are present, the net force on any single charge equals the vector sum of the individual forces from every other charge. You don't need a new formula; you just apply Coulomb's law to each pair and then add the resulting force vectors.

Here's how to work a superposition problem:

1. Identify the charge you're finding the net force on (call it the "target" charge).
2. Calculate the force on the target from each other charge individually using Coulomb's law.
3. Determine the direction of each force (attraction vs. repulsion, along the line connecting the pair).
4. Break each force into x and y components.
5. Sum all x-components and all y-components separately.
6. Use the Pythagorean theorem and inverse tangent to find the magnitude and direction of the net force.

Electric field concept

Field lines representation

Electric field lines give you a visual map of how a positive test charge would move if placed in the field. A few rules for reading them:

• Field lines point away from positive charges and toward negative charges
• The density of lines (how close together they are) indicates field strength: closer lines mean a stronger field
• Field lines never cross each other
• They start on positive charges and end on negative charges (or extend to infinity if there's no nearby opposite charge)

Field strength calculation

The electric field at a point in space is defined as the force per unit positive charge placed at that point:

$E = \frac{F}{q}$

where $E$ is the electric field strength, $F$ is the force on the test charge, and $q$ is the magnitude of the test charge. The units are Newtons per Coulomb (N/C), which are equivalent to Volts per meter (V/m).

The field strength decreases with distance from the source charge, following the same inverse square pattern as the force.

Point charge fields

For a single point charge in a vacuum, the electric field at a distance $r$ is:

$E = k\frac{q}{r^2}$

This field radiates outward symmetrically in all directions from a positive charge (or inward toward a negative charge). The point charge field is the building block for analyzing more complex charge distributions: you can find the total field from multiple charges by applying superposition to the individual fields, just as you do with forces.

Applications of Coulomb's law

Atomic structure

Coulomb's law explains why electrons stay bound to the nucleus: the positive protons attract the negative electrons. The balance between this electrostatic attraction and the quantum mechanical behavior of electrons determines electron configurations, ionization energies, and atomic radii. Chemical bonding between atoms is also fundamentally electrostatic.

Electrostatic devices

Several technologies rely directly on electrostatic forces:

• Electrostatic precipitators remove particulate pollution from industrial exhaust by charging particles and collecting them on oppositely charged plates
• Photocopiers and laser printers use charged drums to attract toner particles in the pattern of an image
• Electrostatic spray painting charges paint droplets so they're attracted to a grounded surface, reducing waste
• Van de Graaff generators accumulate charge to produce high voltages for demonstrations and research

Particle accelerators

Particle accelerators use strong electric fields to push charged particles to extremely high speeds. The same Coulomb's law principles that describe the force between two charges also govern how electric fields accelerate, focus, and steer particle beams. These machines are used in fundamental physics research and in medical applications like radiation therapy.

Limitations and assumptions

Vacuum vs. dielectric media

Coulomb's law in its standard form assumes the charges are in a vacuum (or approximately, in air). When a dielectric material (an insulator like glass, water, or plastic) fills the space between charges, the force is reduced. The material's dielectric constant $\kappa$ (also called relative permittivity) modifies the equation:

$F = \frac{1}{\kappa} \cdot k\frac{q_1 q_2}{r^2}$

Since $\kappa > 1$ for all materials, the force in a dielectric is always weaker than in a vacuum.

Point charge approximation

Coulomb's law treats charges as if they're concentrated at a single point. This works well when the distance between objects is much larger than their size. When charged objects are close together or have significant spatial extent, you need to account for the actual charge distribution, which requires integration techniques covered later in the course.

Static charge distribution

The law assumes charges are stationary. It doesn't account for effects that arise when charges move (which produces magnetic fields) or when charge distributions change over time. For moving charges, you'll eventually need the full framework of electromagnetism.

Historical context

Coulomb's experiments

Charles-Augustin de Coulomb established the inverse square law for electrostatic force in the 1780s. He used a torsion balance, a device with a lightweight bar suspended by a thin fiber. By measuring how much the fiber twisted when charged objects were brought near each other, he could precisely determine the relationship between force, charge, and distance.

Developments in electrostatics

Coulomb's work built on earlier contributions from scientists like Benjamin Franklin (who established the convention of positive and negative charge) and the inventors of early charge-storing devices like the Leyden jar. These developments collectively laid the groundwork for the study of electric currents and, eventually, the full theory of electromagnetism.

Modern interpretations

At the quantum level, electrostatic interactions are described by quantum electrodynamics (QED), which treats the electromagnetic force as an exchange of virtual photons. For the purposes of this course, the classical Coulomb's law is accurate and sufficient for the problems you'll encounter.

Comparison with other forces

Gravitational force analogy

Coulomb's law and Newton's law of gravitation have strikingly similar forms: both are inverse square laws. But there are key differences:

• Gravity is always attractive; the electrostatic force can be attractive or repulsive
• The electrostatic force is enormously stronger. Between a proton and an electron, the electrostatic force is roughly $10^{39}$ times stronger than the gravitational force
• Both forces have infinite range

This analogy is useful for building intuition, but don't forget that the ability to both attract and repel is what makes electrostatics so much richer in behavior.

Electromagnetic unification

The electrostatic force is one piece of the broader electromagnetic interaction. In the 19th century, James Clerk Maxwell unified electricity and magnetism into a single framework (Maxwell's equations), showing that electric and magnetic fields are deeply interconnected and that light itself is an electromagnetic wave.

Strong vs. weak interactions

The electrostatic force is one of the four fundamental forces of nature. For comparison:

• The strong nuclear force holds quarks together inside protons and neutrons and binds nuclei; it's the strongest force but acts only over very short ranges (about $10^{-15}$ m)
• The weak nuclear force is responsible for certain types of radioactive decay
• The electromagnetic force (which includes Coulomb's law) dominates at atomic and molecular scales
• Gravity is by far the weakest but dominates at astronomical scales because it's always attractive and mass adds up

Experimental verification

Torsion balance method

Coulomb's original torsion balance remains a classic demonstration of how to measure small forces. A charged sphere on a suspended bar twists toward or away from a second charged sphere. The angle of twist is proportional to the force, allowing precise measurements that confirmed the inverse square relationship.

Modern measurement techniques

Today, instruments like atomic force microscopes and electrostatic force microscopes can measure electrostatic forces at the nanoscale. High-precision capacitance bridges and computer-controlled data acquisition systems allow measurements far more accurate than anything Coulomb could achieve.

Precision and accuracy considerations

Accurate electrostatic measurements require careful control of environmental factors. Humidity can cause charge to leak away, and stray electromagnetic fields can interfere with readings. Experimenters use shielding, controlled environments, and statistical analysis to minimize these errors.

Mathematical analysis

Inverse square relationship

The force drops off as the square of the distance:

$F \propto \frac{1}{r^2}$

This means if you triple the distance between two charges, the force becomes $\frac{1}{9}$ of its original value. This rapid falloff is shared by other physical phenomena, including gravitational force and light intensity.

Vector components

To find the force between charges that aren't aligned along a single axis:

1. Draw a diagram showing the positions of the charges.
2. Find the distance $r$ between them using the distance formula.
3. Calculate the magnitude of the force using Coulomb's law.
4. Determine the angle of the force vector relative to your coordinate axes.
5. Resolve into components: $F_x = F\cos\theta$ and $F_y = F\cos\theta$ (use $\sin$ and $\cos$ appropriately based on your angle reference).

Multiple charge systems

For systems with three or more charges, apply superposition:

• Calculate the force from each source charge on the target charge separately
• Resolve each force into components
• Add all x-components together and all y-components together
• Combine to find the net force magnitude and direction

These problems can get involved, but the underlying physics is always the same: Coulomb's law applied pairwise, then vector addition.

Units and constants

SI units for charge and force

• Charge: Coulombs (C). One Coulomb is a huge amount of charge; typical static electricity involves microcoulombs ($\mu\text{C} = 10^{-6} \text{ C}$) or nanocoulombs ($\text{nC} = 10^{-9} \text{ C}$)
• Force: Newtons (N)
• Distance: meters (m)
• Electric field: N/C or equivalently V/m

Coulomb constant

The Coulomb constant $k$ has a value of approximately $8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$. It's a fundamental constant in electromagnetism, related to the speed of light and the magnetic permeability of free space.

Permittivity of free space

The permittivity of free space $\varepsilon_0$ measures a vacuum's ability to support electric fields. Its value is approximately $8.85 \times 10^{-12} \text{ F/m}$ (farads per meter). It connects to the Coulomb constant through:

$k = \frac{1}{4\pi\varepsilon_0}$

You'll see Coulomb's law written both ways in textbooks. They're equivalent:

$F = k\frac{q_1 q_2}{r^2} = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^2}$

Microscopic vs. macroscopic effects

Atomic level interactions

At the atomic scale, Coulomb's law governs electron behavior in orbitals, the strength of ionic and covalent bonds, and the electronic properties of materials. Quantum mechanics modifies the picture, but the underlying force is still electrostatic.

Bulk material properties

On a larger scale, electrostatic interactions determine a material's conductivity, dielectric properties, and capacitance. The mechanical hardness of ionic crystals (like table salt) comes from the strong electrostatic bonds between positive and negative ions arranged in a lattice.

Screening and shielding

In conductors and electrolytes, mobile charge carriers rearrange themselves to partially cancel external electric fields. This screening effect reduces the range of electrostatic interactions inside the material. In a metal conductor, free electrons redistribute so that the electric field inside is zero in electrostatic equilibrium. In plasmas and salt solutions, screening limits how far the influence of a single charge extends.