Field of a charged sphere and potential

Inside a conducting sphere of radius R and charge q spread evenly over its surface, the field strength is zero by superposition: E = 0 (r < R). Outside it (r ≥ R) the field equals that of a point charge at the centre: E = kq/r²; on the surface E = kq/R², and it falls with distance as 1/r². A dielectric has very few free charges; in an external field its atoms and molecules are polarised and create their own field E′ opposed to the external E₀: E = E₀ – E′. The factor by which the field is weakened is the dielectric permittivity ε = E₀/E (dimensionless; 1 in vacuum, ≈ 1 in air, ≈ 2–3 in oil, ≈ 81 in water). The field of a point charge in a dielectric is E = kq/(εr²), and the force between charges is F = kq₁q₂/(εr²), ε times smaller than in vacuum.