Derive Gauss law in integral form.

To derive Gauss's law in integral form, also known as Gauss's theorem, follow these steps:

Step 1: Start with the divergence theorem:
The divergence theorem relates the flux of a vector field through a closed surface to the divergence of the field within the enclosed volume. It is given by:
∫∫S F · dA = ∫∫∫V (∇ · F) dV,
where S is a closed surface that encloses a volume V, F is a vector field, dA is a differential surface vector pointing outward, and dV is a differential volume element.

Step 2: Apply the divergence theorem to the electric field:
Replace the vector field F with the electric field E in the divergence theorem equation. This gives us:
∫∫S E · dA = ∫∫∫V (∇ · E) dV.

Step 3: Use the definition of electric flux:
The electric flux Φ through a closed surface S is defined as the integral of the dot product of the electric field E and the differential surface vector dA:
Φ = ∫∫S E · dA.

Step 4: Apply the definition of electric flux to the previous equation:
By substituting the definition of electric flux into the equation from Step 2, we get:
Φ = ∫∫∫V (∇ · E) dV.

Step 5: Recognize that Φ is independent of the volume:
The electric flux through a closed surface does not depend on the size or shape of the volume enclosed. Therefore, we can remove the integral over V, leaving us with:
Φ = ∫∫S E · dA = Q,
where Q is the total charge enclosed by the closed surface.

Step 6: Apply Gauss's law in integral form:
Finally, we arrive at Gauss's law in integral form:
∫∫S E · dA = Q/ε0,
where Q is the total charge enclosed by a closed surface S, E is the electric field, and ε0 is the permittivity of free space.

This derivation shows the relationship between the electric flux through a closed surface and the charge enclosed by that surface. It forms the basis of understanding how electric fields are generated by charges.