Coulomb's Law (DP IB Physics: HL): Revision Note

Coulomb's Law

• All charged particles generate an electric field

  • This field exerts a force on charged particles which are nearby

• The electric force between two charges is defined by Coulomb’s law, which states that:

The electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of their separation

• This electric force can be calculated using the expression:

$F=k\frac{q_1{q}_2}{r^2}$

• Where:

  • F = electric force (N)

  • q₁, q₂ = magnitudes of the charges (C)

  • r = distance between the centres of the two charges (m)

  • k = Coulomb constant (8.99 × 10⁹ N m² C^–2)

• Coulomb's law for two charges is analogous to Newton's law of gravitation for two masses

  • This means that electric and gravitational forces are very similar

  • For example, both forces follow an inverse square law with the separation between charge or mass

Electrostatic attraction between two charges

The attractive electric force F between two point charges +q₁ and −q₂ with a separation of r is defined by Coulomb’s law

• Coulomb's constant is given by:

$k=\frac{1}{4\mathrm{\pi }{\epsilon }_{\mathit{0}}}$

• Where ε₀ is the permittivity of free space

  • ε₀ = 8.85 × 10^–12 C² N^–1 m^–2 and refers to charges in a vacuum

  • The value of the permittivity of air is taken to be the same as ε₀

  • All other materials have a higher permittivity ε > ε₀

  • ε is a measure of the resistance offered by a material in creating an electric field within it

• The value of k depends on the material between the charges

  • In a vacuum, k = 8.99 × 10⁹ N m² C^–2

Repulsive & Attractive Forces 

• Unlike the gravitational force between two masses which is only attractive, electric forces can be attractive or repulsive

• Between two charges of the same type: 

  • The product q₁q₂ is positive, so the forces have positive signs

  • Positive forces mean the charges experience repulsion

• For two opposite charges:

  • The product q₁q₂ is negative, so the forces have negative signs

  • Negative forces mean the charges experience attraction

Different Values of Permittivity

• Permittivity is the measure of how easily an electric field can be generated in a certain material

• The relative permittivity ε_r is sometimes known as the dielectric constant

• For a given material, it is defined as:

The ratio of the permittivity of a material to the permittivity of free space

• Relative permittivity can be expressed as:

${\epsilon }_r=\frac{\epsilon }{{\epsilon }_0}$

• Where:

  • ε_r = relative permittivity

  • ε = permittivity of a material (F m⁻¹)

  • ε₀ = permittivity of free space (F m⁻¹)

• Relative permittivity has no units because it is a ratio of two values with the same unit

• When there is a material between two charges, the Coulomb constant becomes

$k=\frac{1}{4\mathrm{\pi }\epsilon }$

• In air, the relative permittivity is 1, so $\epsilon ={\epsilon }_0$

• In other materials, the Coulomb constant reduces as $\epsilon ={\epsilon }_r{\epsilon }_0$

Examples of relative permittivity

• Some values of relative permittivity for different insulators are shown in the table below: