Physics (9630) | Coulomb’s law

Welcome to Electric Fields: Understanding Coulomb's Law

Hello! This chapter is your foundation for understanding electricity at a deeper level. Just like Newton's Law of Gravitation explains how masses attract each other, Coulomb’s Law explains the force between electric charges.
It is the starting point for the entire concept of electric fields and capacitance. Don't worry if the formulas look complicated; we will break them down step-by-step!

1. What is Electric Charge? (A Quick Review)

Before tackling the forces, let's briefly recall the concept of charge, the fundamental property that causes this interaction:

• Types of Charge: There are two types: Positive (+) and Negative (-).
• Basic Rule of Interaction:
  Like charges repel (e.g., + repels +, or - repels -).
  Opposite charges attract (e.g., + attracts -).
• Unit of Charge: The SI unit for electric charge is the Coulomb (C). One coulomb is a huge amount of charge!

Key Takeaway: Electric force is what dictates whether charges push apart or pull together. Coulomb's Law tells us exactly how strong that push or pull is.

2. The Core Concept: Coulomb's Law (3.8.1)

Coulomb's Law describes the magnitude and direction of the electric force (electrostatic force) between two stationary, electrically charged particles (called point charges).

Key Principles of Coulomb’s Law

The force \(F\) between two point charges, \(Q_1\) and \(Q_2\), depends on two main factors:

1. The magnitude of the force is directly proportional to the product of the charges (\(F \propto Q_1 Q_2\)).

   Analogy: If you double one charge, the force doubles. If you double both charges, the force quadruples.

2. The magnitude of the force is inversely proportional to the square of the distance, \(r\), separating them (\(F \propto \frac{1}{r^2}\)).

   This is known as the Inverse Square Law. If you double the distance between the charges, the force drops to a quarter of its original value.

3. The Mathematical Formula

Combining these principles, the formula for the electrostatic force \(F\) between two point charges \(Q_1\) and \(Q_2\) separated by a distance \(r\) in a vacuum is:

$$F = \frac{1}{4\pi\varepsilon_0} \frac{Q_1 Q_2}{r^2}$$

Breaking Down the Equation

a) The Variables:

• \(F\): The magnitude of the electrostatic force (measured in Newtons, N).
• \(Q_1\) and \(Q_2\): The magnitudes of the two point charges (measured in Coulombs, C).
• \(r\): The distance between the centres of the charges (measured in metres, m).

b) The Coulomb Constant ($k$):

The term \(\frac{1}{4\pi\varepsilon_0}\) is often simplified and replaced by the Coulomb Constant, \(k\).

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

In examination settings, you will often use the numerical value for \(k\):
\(k \approx 8.99 \times 10^9 \, \text{N m}^2 \text{ C}^{-2}\)

c) Permittivity of Free Space (\(\varepsilon_0\)):

The term \(\varepsilon_0\) (epsilon naught) is the permittivity of free space (or vacuum).

• It is a constant that represents how easily an electric field can pass through a vacuum.
• Its value is approximately: \(\varepsilon_0 \approx 8.85 \times 10^{-12} \, \text{F m}^{-1}\) (Farads per metre, though you primarily need to know its role in the equation).

4. The Vector Nature of Electrostatic Force

Force is a vector quantity, meaning it has both magnitude and direction. Coulomb's Law gives us the magnitude, but we determine the direction based on the charges:

• Repulsion: If the charges are the same sign (both positive or both negative), the force acts outward along the line connecting the charges (pushing them apart).
• Attraction: If the charges are opposite signs (one positive, one negative), the force acts inward along the line connecting the charges (pulling them together).

Tip for Calculations: When calculating the magnitude of $F$, you usually substitute only the positive values of \(Q_1\) and \(Q_2\). You then determine the direction (attraction or repulsion) separately based on the signs.

4.1 Point Charges and Charged Spheres

Coulomb's Law is strictly defined for point charges (charges concentrated at a single geometric point). However, the syllabus allows us to make two important approximations:

1. Air as a Vacuum: When calculating the force between charges in air, you can treat air as if it were a vacuum. Therefore, you continue to use the vacuum permittivity, \(\varepsilon_0\).
2. Charged Spheres: For a uniformly charged sphere, or any sphere where the charges are stationary, you can model the sphere as a single point charge located at its centre. This simplifies problems significantly, as the distance \(r\) is measured centre-to-centre.

5. Comparing Electric Force and Gravitational Force

The syllabus requires you to compare the magnitude and nature of electrostatic forces and gravitational forces, particularly at the subatomic level.

Similarities:

Both forces share the same mathematical structure:

• Both are inverse square laws (\(F \propto 1/r^2\)).
• Both depend on the product of the interacting properties (masses \(m_1 m_2\) for gravity, charges \(Q_1 Q_2\) for electrostatics).

Differences:

Feature	Electrostatic Force (\(F_E\))	Gravitational Force (\(F_G\))
Mediated by	Electric Charge (\(Q\))	Mass (\(m\))
Nature	Can be Attractive or Repulsive.	Only Attractive.
Strength	Extremely Strong.	Extremely Weak.
Range	Infinite range.	Infinite range.

Magnitude Comparison (The Crucial Difference)

The most striking difference is the immense relative strength of the electric force compared to gravity.

Did you know? The force holding an electron to a proton in a hydrogen atom (electrostatic force) is about \(10^{39}\) times stronger than the gravitational force between them!

For subatomic particles (which have both mass and charge), the gravitational force is generally negligible compared to the electrostatic force. This is why gravitational effects are ignored in most particle physics problems.

The reason gravity seems strong in everyday life is that charges typically cancel out (objects are neutral), while mass is always positive and accumulates, making gravity noticeable only with very large masses (like planets).

6. Summary of Key Terms and Concepts

Important Definitions:

• Point Charge: An idealised particle with charge concentrated at a single point.
• Coulomb's Law: Describes the electrostatic force between two point charges.
• Permittivity of Free Space (\(\varepsilon_0\)): A physical constant reflecting the ability of a vacuum to permit electric fields (\(\approx 8.85 \times 10^{-12} \, \text{F m}^{-1}\)).
• Inverse Square Law: The force is proportional to \(1/r^2\).

Formula Check:

$$F = \frac{1}{4\pi\varepsilon_0} \frac{Q_1 Q_2}{r^2} = k \frac{Q_1 Q_2}{r^2}$$

Keep Practicing! Coulomb’s Law is fundamental. Master the inverse square relationship and the application to point charges and spheres, and you’ll have a fantastic start to this section!