Effect of dielectric on capacitance

With vacuum between the plates, E₀ = σ/ε₀, V₀ = E₀d and C₀ = ε₀A/d.

A dielectric filling the gap is polarised; its bound surface charges ±σₚ cut the effective charge density to σ − σₚ, so E = (σ − σₚ)/ε₀.

For a linear dielectric σ − σₚ = σ/K, where K > 1 is the dielectric constant. The field and V both fall by the factor K for the same charge.

Hence C = Kε₀A/d = KC₀. The dielectric constant is the factor by which capacitance grows when the dielectric fills the space fully.

Permittivity of the medium ε = ε₀K; K = ε/ε₀ is dimensionless. For vacuum K = 1.

C = KC₀ holds for a capacitor of any shape, so it can serve as the definition of K.

Slab partly filling the gap: a slab of constant K and thickness 3d/4 (full plate area) makes V = E₀d(K + 3)/(4K) for the same charge, so C = 4KC₀/(K + 3).