Which equation expresses the relation between the divergence of the electric field and charge density?

Which equation expresses the relation between the divergence of the electric field and charge density?

• A. ∇·E = ρ/ε0
• B. ∇×B = μ0 J + μ0 ε0 ∂E/∂t
• C. ∇·B = 0
• D. ∇×E = -∂B/∂t

Answer: (A)

Gauss's law in differential form shows how charges act as sources of the electric field, linking the divergence of E to charge density. The divergence operator ∇·E measures how much the field lines spread out from a point—essentially the net outward flux per unit volume. According to Gauss's law, this local spreading is proportional to the charge density ρ, with the vacuum permittivity ε0 setting the units: ∇·E = ρ/ε0. So, when there is charge at a point, the field has a source there and the divergence is nonzero; in regions with no charge, the divergence is zero. The other equations describe different aspects of electromagnetic fields. ∇×B relates to current and the time rate of change of E (Ampère–Maxwell law), ∇·B = 0 expresses that magnetic monopoles don’t exist, and ∇×E = -∂B/∂t is Faraday’s law showing how changing magnetic fields induce circulating electric fields.

Gauss's law in differential form shows how charges act as sources of the electric field, linking the divergence of E to charge density. The divergence operator ∇·E measures how much the field lines spread out from a point—essentially the net outward flux per unit volume. According to Gauss's law, this local spreading is proportional to the charge density ρ, with the vacuum permittivity ε0 setting the units: ∇·E = ρ/ε0. So, when there is charge at a point, the field has a source there and the divergence is nonzero; in regions with no charge, the divergence is zero.

The other equations describe different aspects of electromagnetic fields. ∇×B relates to current and the time rate of change of E (Ampère–Maxwell law), ∇·B = 0 expresses that magnetic monopoles don’t exist, and ∇×E = -∂B/∂t is Faraday’s law showing how changing magnetic fields induce circulating electric fields.