Electric Force Between Point Charges: Coulomb's Law

Hey there! Welcome to one of the most fundamental topics in electricity: understanding the force that holds charged particles together—or pushes them apart! This chapter is all about quantifying this force using Coulomb's Law. Don't worry if equations look daunting; we’ll break down every symbol and explain exactly how these tiny forces govern the world around us, from static cling to advanced electronics.

Mastering this concept is vital because it is the foundation for understanding electric fields and electric potential later in the syllabus!

1. The Basics of Electric Charge (A Quick Refresher)

Before we calculate the force, let’s quickly recall what we are dealing with: electric charge.

  • There are two types of charge: positive (+) and negative (-).
  • Key Rule: Opposite charges attract (like protons and electrons), and like charges repel (like two electrons).
  • The standard SI unit for charge is the Coulomb (C). One Coulomb is a very large amount of charge!

Analogy: Think of charges like magnetic poles, but instead of North and South, we have Positive and Negative. The forces act along the line connecting the centers of the charges.

2. Introducing Coulomb’s Law

In the late 18th century, Charles-Augustin de Coulomb quantified the electrostatic force. This resulting principle, known as Coulomb's Law, tells us exactly how strong the electric force is between two point charges.

The law states that the force $F$ between two point charges is:

  • Directly proportional to the product of the magnitudes of the charges (\(Q_1 \times Q_2\)). This means if you double one charge, the force doubles.
  • Inversely proportional to the square of the distance between them (\(r^2\)). This is the famous Inverse Square Law.

Did You Know? Coulomb's Law has the exact same mathematical structure as Newton's Law of Gravitation. The key difference is that gravity is always attractive, while the electric force can be attractive or repulsive!

Key Takeaway:

The electric force gets much weaker, very quickly, as you move the charges apart. If you triple the distance (\(r \rightarrow 3r\)), the force decreases by a factor of nine (\(1/3^2\)).

3. The Full Equation for Coulomb’s Law (Syllabus Requirement 18.3)

The electric force \(F\) acting between two point charges \(Q_1\) and \(Q_2\) separated by a distance \(r\) in free space is given by:

$$F = \frac{Q_1 Q_2}{4\pi \epsilon_0 r^2}$$
Breaking Down the Components:

Let's make sure we understand every part of this powerful equation:

  • \(F\): The magnitude of the electric force (measured in Newtons, N).
  • \(Q_1\) and \(Q_2\): The magnitudes of the two charges (measured in Coulombs, C).
  • \(r\): The distance between the centers of the two charges (measured in metres, m). Remember, it must be squared!
  • \(\epsilon_0\) (Epsilon naught): This is the permittivity of free space.
    • It is a physical constant that measures how easily an electric field can pass through a vacuum (or air, which is usually considered 'free space' in these calculations).
    • The value provided in the data sheet is usually \(\epsilon_0 \approx 8.85 \times 10^{-12} \text{ F m}^{-1}\) (Farads per metre).

The Constant \(k\):
Sometimes, physicists group the constants together. The term \(\frac{1}{4\pi \epsilon_0}\) is often called the Coulomb constant, \(k\).
$$k = \frac{1}{4\pi \epsilon_0} \approx 9.0 \times 10^9 \text{ N m}^2 \text{ C}^{-2}$$ Using \(k\) makes the equation simpler to write: \(F = k \frac{Q_1 Q_2}{r^2}\). However, for your Cambridge exam, be prepared to use the version with \(\epsilon_0\) as given in the formula sheet.

A Note on Force Direction (Vectors!)

Remember that force \(F\) is a vector quantity—it has both magnitude and direction.

  • When using the formula to find the magnitude, you generally only substitute the absolute values of \(Q_1\) and \(Q_2\) (i.e., ignore the negative signs).
  • To determine the direction of the force, you rely on the fundamental rule: attraction (towards each other) if charges are opposite, and repulsion (away from each other) if charges are the same.

🚨 Quick Review Box: Common Mistakes to Avoid

1. Forgetting \(r^2\): Always square the distance \(r\). This is the most frequent error.

2. Units: Ensure charges are in Coulombs (C) and distances are in metres (m) before calculating. If you see micro-Coulombs (\(\mu\text{C}\)) or centimetres (cm), convert them using prefixes first!

3. Signs: Do not put negative signs for charge into the equation unless explicitly required by a specific problem setup (which is rare in 9702). Use the signs only to decide if the force is attractive or repulsive.

4. Dealing with Real-World Charges: The Spherical Approximation

The definition of Coulomb's Law relies on point charges—idealized objects with charge concentrated at a single point.

In reality, charges are often distributed over objects, like a small metal sphere. The syllabus requires you to know how to handle these situations:

Rule for Spherical Conductors (Syllabus 18.3, point 1)

For a point located outside a spherically shaped conductor (like a charged metal ball), we can simplify the problem significantly:

The entire charge \(Q\) on the sphere can be considered to be concentrated as a point charge at its centre.

Why does this simplification work? Due to the symmetry of the sphere, the electric field lines behave exactly as if all the charge were squashed right into the middle point. This is the same principle you use when calculating gravitational forces exerted by planets.

Step-by-Step for Spheres:

  1. Identify the separation distance \(r\). This distance must be measured from the center of the charged sphere to the location of the other charge.
  2. Treat the sphere as a point charge at that central position.
  3. Apply Coulomb's Law using the total charge of the sphere (\(Q_1\)) and the external charge (\(Q_2\)).
Key Takeaway:

When calculating the electric force exerted by a uniform spherical charge distribution on an external point charge, always measure the distance \(r\) from the centre of the sphere.

Chapter Summary: Electric Force

Here are the core concepts you must recall from this section:

  • Fundamental Force: Electric force is caused by electric charges. Like charges repel, opposite charges attract.
  • Coulomb's Law Equation: The magnitude of the force \(F\) is calculated using \(F = \frac{Q_1 Q_2}{4\pi \epsilon_0 r^2}\).
  • Inverse Square Relationship: The force \(F\) is proportional to \(1/r^2\).
  • Permittivity: \(\epsilon_0\) is the permittivity of free space, allowing us to calculate the force in a vacuum or air.
  • Approximation: For a point outside a uniformly charged spherical conductor, treat the conductor's charge as a point charge located at its geometric centre.

Congratulations, you’ve grasped the mathematics behind electrostatic forces! This is a massive step towards understanding electric fields!