18.2 Coulomb's law

Coulomb's law describes the force between charged particles, playing the same role in electrostatics that Newton's law of gravitation plays for masses. The key difference: electric force can attract or repel, depending on the signs of the charges involved.

Coulomb's Law and Electrostatic Forces

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Coulomb's law calculations

Coulomb's law gives you the magnitude of the electrostatic force between two charged objects:

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

Here's what each variable means:

• $F$ is the magnitude of the electrostatic force, in newtons (N)
• $k$ is Coulomb's constant: $8.99 \times 10^9 \frac{N \cdot m^2}{C^2}$ (in a vacuum)
• $|q_1|$ and $|q_2|$ are the absolute values of the two charges, in coulombs (C)
• $r$ is the distance between the centers of the two charged objects, in meters (m)

The force always acts along the straight line connecting the two charges. Its direction depends on the signs:

• Like charges (positive-positive or negative-negative) repel each other
• Opposite charges (positive-negative) attract each other

Because force is a vector, you need to track both magnitude (from the equation) and direction (from the charge signs) when solving problems.

Factors affecting electrostatic force

The equation tells you exactly how force responds to changes in charge and distance, and these proportional reasoning questions show up constantly on tests.

Charge dependence (direct proportion):

• The force is directly proportional to the product $|q_1||q_2|$
• Doubling one charge doubles the force
• Doubling both charges quadruples the force

Distance dependence (inverse-square):

• The force is inversely proportional to $r^2$
• Doubling the distance cuts the force to $\frac{1}{4}$ of its original value
• Tripling the distance cuts the force to $\frac{1}{9}$

A quick way to handle these problems: set up a ratio. If you double $r$, the new force is $F_{new} = \frac{F_{old}}{2^2} = \frac{F_{old}}{4}$. You don't need to recalculate everything from scratch.

The permittivity of the medium also matters. In materials other than a vacuum, the effective value of $k$ changes, which weakens the force compared to vacuum conditions.

Coulomb's law vs Newton's gravitation

These two laws have strikingly similar mathematical forms, and comparing them is a great way to understand both more deeply.

Similarities:

• Both are action-at-a-distance forces (no physical contact needed)
• Both follow an inverse-square relationship with distance
• Both act along the line connecting the two objects' centers

Differences:

Electric Fields and Superposition

An electric field is the region around a charged object where another charge would experience a force. Think of it as the "zone of influence" that a charge creates in the space around it.

When multiple charges are present, the superposition principle applies: the total electric field at any point is the vector sum of the individual fields from each charge. You calculate each field separately, then add them as vectors (accounting for direction).

Field lines provide a visual way to map electric fields. The lines point in the direction a positive test charge would be pushed, and they're drawn closer together where the field is stronger. Field lines point away from positive charges and toward negative charges.