State Gauss's law in differential form.

State Gauss's law in differential form.

• A. ∇ × E = ρ / ε0
• B. ∇ · E = ρ / ε0
• C. ∇² E = ρ / ε0
• D. ∇ · E = ε0 / ρ

Answer: (B)

The idea being tested is how Gauss's law ties the local behavior of the electric field to the charge density at a point. The divergence operator ∇·E measures how much the field “diverges” from a small volume—that is, how many field lines are exiting per unit volume. Gauss’s law in differential form states that this local divergence equals the charge density at that point divided by the vacuum permittivity: ∇·E = ρ/ε0. This is the pointwise version of the total-flux relation ∮ E·dA = Q_enclosed/ε0, obtained by applying the divergence theorem. Why this form is correct: it mirrors the intuitive idea that more charge nearby causes a stronger outward flux of the electric field, and the factor 1/ε0 sets the scale between field flux and charge. The divergence of E directly corresponds to how much field lines are emanating per unit volume, which matches the presence of charge there. Why the other forms aren’t correct: the curl of E, ∇×E, is tied to changing magnetic fields (Faraday’s law) and not to charge density. The Laplacian, ∇²E, is not the fundamental relation for Gauss’s law; it appears in Poisson’s equation for the potential as ∇²V = -ρ/ε0 with E = -∇V, but Gauss’s law itself is about ∇·E. And the expression with ε0/ρ has incorrect units and wrong physical meaning.

The idea being tested is how Gauss's law ties the local behavior of the electric field to the charge density at a point. The divergence operator ∇·E measures how much the field “diverges” from a small volume—that is, how many field lines are exiting per unit volume. Gauss’s law in differential form states that this local divergence equals the charge density at that point divided by the vacuum permittivity: ∇·E = ρ/ε0. This is the pointwise version of the total-flux relation ∮ E·dA = Q_enclosed/ε0, obtained by applying the divergence theorem.

Why this form is correct: it mirrors the intuitive idea that more charge nearby causes a stronger outward flux of the electric field, and the factor 1/ε0 sets the scale between field flux and charge. The divergence of E directly corresponds to how much field lines are emanating per unit volume, which matches the presence of charge there.

Why the other forms aren’t correct: the curl of E, ∇×E, is tied to changing magnetic fields (Faraday’s law) and not to charge density. The Laplacian, ∇²E, is not the fundamental relation for Gauss’s law; it appears in Poisson’s equation for the potential as ∇²V = -ρ/ε0 with E = -∇V, but Gauss’s law itself is about ∇·E. And the expression with ε0/ρ has incorrect units and wrong physical meaning.