Electrostatics

Background and naming

This is Coulomb's Law, governing the magnitude of the electric force between two charged particles. Named after Charles Augustine de Coulomb, who published experimental determinations in 1785.

The force on an object due to another charged object is denoted as F_e (magnitude of the electric force).

Sometimes called the electric force, Coulomb's force, or the electrostatic force.

Magnitude of the electric force

The magnitude of the electric force between two point charges is given by

Fe = k\frac{q1 q_2}{r^2}

where

F_e is the magnitude of the electric force,

k is Coulomb's constant,

q1 and q2 are the charges,

r is the distance between the centers of charge of the two objects.

Important note: in this magnitude form, the sign of the product q1 q2 is not the magnitude; it indicates whether the force is attractive or repulsive in the scalar convention discussed below.

r: distance vs radius

In Coulomb's Law, r is the distance between the centers of charge of the two objects, not the radius.

This is similar but not identical to Newton's law of gravitation, where the distance is the distance between the centers of mass of the two objects.

Potential confusion: sometimes people use r as a radius; the correct interpretation here is the center-to-center distance for charges.

Constants and their scales

Coulomb constant: k = 8.99\times 10^{9}\ \text{N m}^2/\text{C}^2

Universal gravitational constant: G = 6.67\times 10^{-11}\ \text{N m}^2/\text{kg}^2

Relative size: \frac{k}{G} = \frac{8.99\times 10^{9}}{6.67\times 10^{-11}} \approx 1.35\times 10^{20}

Conclusion: the Coulomb constant is enormous compared to the gravitational constant by about a factor of 1.35\times 10^{20}, which helps explain why electric forces often dominate gravitational effects at small scales.

Electric force vs gravity (intuition)

Example: rubbing a balloon on hair gives the balloon a net negative charge and the hair a net positive charge.

The electric attraction/repulsion between charged objects is typically far stronger than gravity for small objects, so the balloon can stick to hair despite gravity pulling downward.

Signs and direction (conventions in the transcript)

The basic magnitude formula yields a nonnegative magnitude, but the sign of the force in the commonly taught scalar form indicates attraction vs repulsion depending on the sign of the charge product:

If q1 q2 > 0 (same signs): force is repulsive.

If q1 q2 < 0 (opposite signs): force is attractive.

In the worked example, with opposite charges, the computed scalar value from the form Fe = k q1 q_2 / r^2 is negative, indicating attraction in that sign convention.

A practical takeaway: negative in this scalar form typically means attraction; positive means repulsion. It is important to interpret the sign in the context of the chosen convention and the physical arrangement.

The vector form clarifies direction more explicitly (see below). The sign convention is a teaching aid to connect algebra with physical behavior (attraction vs repulsion).

Point charges and prefixes for charge units

A point charge is an idealized charge with zero size but finite charge, used in Coulomb's Law.

Prefixes to know:

micro (µ) means 10^{-6}

1\,\mu\text{C} = 1.0\times 10^{-6}\,\text{C}

nano (n) means 10^{-9}

1\,\text{nC} = 1.0\times 10^{-9}\,\text{C}

pico (p) means 10^{-12}

1\,\text{pC} = 1.0\times 10^{-12}\,\text{C}

In the example, charges are given as \pm 5.0\ \mu\text{C} (microcoulombs).

Worked example: two equal but opposite point charges

Given: two equal magnitude point charges, 2.0 m apart, magnitudes \pm 5.0\ \mu\text{C} each, with opposite signs.

Data:

q_1 = +5.0\ \mu\text{C} = +5.0\times 10^{-6}\ \text{C}

q_2 = -5.0\ \mu\text{C} = -5.0\times 10^{-6}\ \text{C}

r = 2.0\ \text{m}

Calculation (magnitude):

Convert charges to coulombs: as above.

Use magnitude form:

Fe = k\frac{|q1 q_2|}{r^2} = (8.99\times 10^{9})\frac{|(+5.0\times 10^{-6})(-5.0\times 10^{-6})|}{(2.0)^2}

= 8.99\times 10^{9} \times \frac{25\times 10^{-12}}{4} = 8.99\times 10^{9} \times 6.25\times 10^{-12}

\approx 5.61875\times 10^{-2}\ \text{N} \approx 0.056\ \text{N}

Sign and interpretation:

The product q1 q2 is negative, so the scalar form gives a negative value in that convention, indicating attraction.

The magnitude is |F_e| \approx 0.056\ \text{N}.

Therefore, the force on each charge due to the other is of magnitude 0.056\ \text{N} directed toward the other charge (i.e., attraction).

Newton's third law consistency: the two forces are equal in magnitude and opposite in direction, forming a force pair.

Vector form and the three common representations

Three common representations of Coulomb’s law discussed: 1) Scalar form with sign (as used above): Fe = k\frac{q1 q_2}{r^2}

Sign indicates attraction (negative) or repulsion (positive) in the chosen convention.

2) Magnitude form: (always nonnegative)

|Fe| = k\frac{|q1 q2|}{r^2} 3) Vector form using a unit vector: \mathbf{F}{12} = k\frac{q1 q2}{r^2}\hat{r}{12}

Here, \hat{r}_{12} is the unit vector directed from charge 1 toward charge 2.

The product q1 q2 still encodes whether the force is attractive or repulsive, but the vector direction is explicitly given by the unit vector.

Note from the transcript: unit-vector form was introduced but not used in the worked examples; students were cautioned to handle sign and direction carefully and to be mindful that the third form provides directional information along the line joining the charges.

Net force with a third charge (conceptual example)

Scenario: Add charge 3 with the same sign as charge 1 on the opposite side from charge 2, both at 2.0 m from charge 1.

Given: charge 1 is +, charge 2 is -, charge 3 is +, all at 2.0 m apart in the described arrangement.

Forces on charge 1:

Force from charge 2 (opposite sign): attractive, magnitude 0.056\ \text{N}, direction toward charge 2.

Force from charge 3 (same sign as charge 1): repulsive, magnitude 0.056\ \text{N}, direction away from charge 3.

In the described arrangement, both forces on charge 1 act in the same direction (to the right), giving a net force of

F_{\text{net}} = 0.056\ \text{N} + 0.056\ \text{N} = 0.112\ \text{N} \approx 0.11\ \text{N}

This net force is directed to the right in the pictured configuration.

Key takeaway: adding a third charge changes the net force through vector addition of individual Coulomb forces; directions depend on relative positions and signs.

Practical takeaways and problem-solving tips

Always check unit consistency: convert micro-, nano-, pico- prefixes to coulombs before plugging into the formula.

Use the magnitude form to avoid sign confusion when you only need the strength.

Use the sign convention to determine attraction vs repulsion in the scalar form, or switch to the vector form for explicit direction.

Remember Newton's third law: the pair of forces on the two charges are equal in magnitude and opposite in direction.

When interpreting results from calculators, don’t rely on the sign alone for direction; instead relate the sign to attraction/repulsion and/or examine the geometry for the actual direction along the line between charges.

Connections to broader physics

Coulomb's Law mirrors the inverse-square structure of Newton's gravitational law, highlighting a common mathematical form across fundamental forces (electric vs gravitational) with vastly different constants, leading to different relative strengths in typical situations.

The large value of k compared to G explains why electric effects often dominate at microscopic scales while gravity dominates at astronomical scales with neutral matter.

Ethical, philosophical, and practical implications (brief)

Understanding electrostatic forces underpins technologies (electronics, sensors, charge management) and safety considerations when dealing with high-voltage charges.

Conceptually, the idea that forces come in action-reaction pairs (Newton's third law) reflects conservation laws and symmetry in physical interactions.

Quick reference formulas

Electric force magnitude: Fe = k\frac{q1 q_2}{r^2}

Magnitude form (nonnegative): |Fe| = k\frac{|q1 q_2|}{r^2}

Vector form (direction along the line joining charges): \mathbf{F}{12} = k\frac{q1 q2}{r^2}\hat{r}{12}

Unit conversions for charge:

1\,\mu\text{C} = 1.0\times 10^{-6}\,\text{C}

1\,\text{nC} = 1.0\times 10^{-9}\,\text{C}

$$1\,\text{pC} =