Coulomb's law: extended (AQA A-Level Physics): Revision Notes

Coulomb's law: extended

Background to electrostatic forces

Static electricity (or electrostatics) describes the interaction between charged objects. Objects can become electrically charged through friction, which transfers electrons between materials. A negatively charged object will attract a positively charged object, while two objects with the same type of charge will repel each other.

Polarisation occurs when charged objects interact with insulators. Molecules within the insulator develop a slight separation of positive and negative charge, with one end becoming slightly positive and the other slightly negative. This creates a net attractive force between the charged object and the polarised material.

Coulomb's law

In 1785, Charles-Augustin de Coulomb conducted experiments measuring the force between charged objects using a torsional balance. His work established a quantitative relationship between charge, separation, and electrostatic force.

This can be expressed mathematically as:

$F = \frac{kQ_1Q_2}{r^2}$

where:

• F is the electrostatic force in newtons (N)
• Q₁ and Q₂ are the point charges in coulombs (C)
• r is the separation between the charge centres in metres (m)
• k is the constant of proportionality

Permittivity of free space

The constant k can be written as:

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

where ε₀ is called the permittivity of free space, a fundamental constant with the value:

$\varepsilon_0 = 8.85 \times 10^{-12} \text{ F m}^{-1}$

Using this relationship, Coulomb's law is more commonly written as:

$F = \frac{1}{4\pi\varepsilon_0} \cdot \frac{Q_1Q_2}{r^2} = \frac{Q_1Q_2}{4\pi\varepsilon_0 r^2}$

Although this equation is strictly valid only in a vacuum (free space), it can be used for charges in air since the effect of air on the electrostatic force is negligible. In other materials, the force between charges is affected, which must be considered separately.

Point charge approximation

A spherically symmetrical object with uniform charge distribution can be treated as a point charge located at its centre. This simplification is particularly useful when calculating forces between charged spheres in experimental setups or between subatomic particles.

Worked example: force between charged spheres

Application to subatomic particles

Coulomb's law applies at all scales, from macroscopic charged objects to subatomic particles. A particularly important application is calculating the electrostatic force between the proton and electron in a hydrogen atom.

For a hydrogen atom:

• The proton has charge +1.6 × 10⁻¹⁹ C
• The electron has charge −1.6 × 10⁻¹⁹ C
• The diameter of the atom is approximately 0.106 nm, so r = 0.053 nm = 0.053 × 10⁻⁹ m

The electrostatic force between them is:

$F_e = \frac{Q_1Q_2}{4\pi\varepsilon_0 r^2}$

$F_e = \frac{(1.6 \times 10^{-19})^2}{4\pi \times 8.85 \times 10^{-12} \times (0.053 \times 10^{-9})^2}$

$F_e = 8.20 \times 10^{-8} \text{ N}$

Comparison with gravitational force

It is instructive to compare the electrostatic force between the proton and electron with the gravitational force between them. Using Newton's law of gravitation with:

• Proton mass: m_p = 1.67 × 10⁻²⁷ kg
• Electron mass: m_e = 9.11 × 10⁻³¹ kg

The gravitational force is:

$F_g = \frac{Gm_p m_e}{r^2}$

$F_g = \frac{6.67 \times 10^{-11} \times 1.67 \times 10^{-27} \times 9.11 \times 10^{-31}}{(0.053 \times 10^{-9})^2}$

$F_g = 3.61 \times 10^{-47} \text{ N}$

Experimental verification

By measuring the force at different separations, the inverse square relationship (F ∝ 1/r²) can be verified experimentally. The charge on each sphere can be measured using a coulombmeter.

Remember!