The Gauss law employs the Gauss theorem for the calculation of charge density.

The Gauss law is a fundamental principle in electromagnetism that relates the electric field to the distribution of electric charges. It states that the total electric flux through a closed surface is equal to the total charge enclosed by that surface divided by the permittivity of free space (ε0).

The Gauss theorem, also known as Gauss's divergence theorem or Gauss's flux theorem, is a mathematical theorem that relates the flux of a vector field through a closed surface to the divergence of the field within the volume enclosed by that surface. It establishes a relationship between the surface integral of a vector field and the volume integral of its divergence.

Explanation:
The Gauss law can be mathematically represented as:

∮E · dA = Qenc/ε0

where ∮E · dA represents the flux of the electric field E through a closed surface, Qenc represents the total charge enclosed by that surface, and ε0 is the permittivity of free space.

To calculate the charge density, we can rearrange the Gauss law equation as:

Qenc = ∮E · dA * ε0

The charge density (ρ) is defined as the charge per unit volume. By dividing the total charge (Qenc) by the volume (V) enclosed by the surface, we can obtain the charge density:

ρ = Qenc / V

Substituting the value of Qenc from the rearranged Gauss law equation, we get:

ρ = (∮E · dA * ε0) / V

Now, we can apply the Gauss theorem to convert the surface integral to a volume integral:

∮E · dA = ∫∫∫ (∇ · E) dV

where ∇ · E represents the divergence of the electric field.

By substituting this into the equation for charge density, we get:

ρ = (∫∫∫ (∇ · E) dV * ε0) / V

This equation allows us to calculate the charge density by evaluating the divergence of the electric field within the volume enclosed by the surface and dividing it by the volume. Thus, the Gauss theorem is employed in the calculation of charge density in accordance with the Gauss law.