State Gauss's law in integral form.

State Gauss's law in integral form.

• A. ∮ E · dA = Q_enclosed / ε0
• B. ∮ E · dA = ε0 Q_enclosed
• C. ∮ E · dA = ρ / ε0
• D. ∮ E · dA = Q_enclosed / ε

Answer: (A)

The fundamental idea is that the total electric flux through a closed surface is determined only by the charge enclosed by that surface. Gauss's law in integral form expresses this relationship by equating the surface integral of the electric field over the closed surface to the enclosed charge divided by the permittivity of free space. So the correct form is: ∮ E · dA = Q_enclosed / ε0. Here ∮ E · dA represents the net outward flux of the electric field through the closed surface, Q_enclosed is the total charge inside the surface, and ε0 is the permittivity of free space, ensuring the units and proportionality are right. This form makes intuitive sense: if there is no charge inside, the net flux through the surface is zero because field lines entering and leaving balance. If there is positive charge inside, the outward flux is positive, proportional to how much charge is enclosed. The other expressions are not correct for Gauss's law in vacuum: multiplying by ε0 on the right would be dimensionally and conceptually inconsistent; using ρ would involve a density instead of total enclosed charge; and dividing by ε (without the subscript) would imply a medium-specific constant or a different quantity (in general, Gauss's law in vacuum uses ε0).

The fundamental idea is that the total electric flux through a closed surface is determined only by the charge enclosed by that surface. Gauss's law in integral form expresses this relationship by equating the surface integral of the electric field over the closed surface to the enclosed charge divided by the permittivity of free space.

So the correct form is: ∮ E · dA = Q_enclosed / ε0. Here ∮ E · dA represents the net outward flux of the electric field through the closed surface, Q_enclosed is the total charge inside the surface, and ε0 is the permittivity of free space, ensuring the units and proportionality are right.

This form makes intuitive sense: if there is no charge inside, the net flux through the surface is zero because field lines entering and leaving balance. If there is positive charge inside, the outward flux is positive, proportional to how much charge is enclosed.

The other expressions are not correct for Gauss's law in vacuum: multiplying by ε0 on the right would be dimensionally and conceptually inconsistent; using ρ would involve a density instead of total enclosed charge; and dividing by ε (without the subscript) would imply a medium-specific constant or a different quantity (in general, Gauss's law in vacuum uses ε0).