Understanding the Concept:
Let us review the standard four Maxwell Equations in their point/differential form representation:
• Gauss's Law for Electricity: $\nabla \cdot \mathbf{D} = \rho_v$ (Electric field flux density divergence matches volumetric charge density; hence electric fields originate/terminate on charges).
• Gauss's Law for Magnetism: $\nabla \cdot \mathbf{B} = 0$ (Divergence of magnetic flux density is zero everywhere; magnetic lines always form continuous closed loops).
• Faraday's Law of Induction: $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ (A time-varying magnetic flux induces a curled electric field).
• Ampere's Circuital Law (with Maxwell's correction): $\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}$ (Current density and changing displacement fields create a curled magnetic field).
Step-by-step Evaluation of Options:
• Option A evaluation: $\nabla \cdot \mathbf{D} = \rho_v$ establishes that the flux density scale is dictated by the charge density. This is true.
• Option B evaluation: $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ demonstrates that dynamic time modifications to $\mathbf{B}$ generate an electric field. This is true.
• Option C evaluation: In vector calculus, the presence of operational sources and sinks inside a vector field is indicated by a non-zero value for the divergence ($\nabla \cdot \mathbf{A} \neq 0$). Ampere's Law describes the curl ($\nabla \times \mathbf{H}$) of a magnetic field, which represents rotational circulation around an axis, not sources or sinks. Furthermore, because $\nabla \cdot \mathbf{B} = 0$, magnetic fields possess no sources or sinks. Therefore, this statement is false, making it the correct option selection.
• Option D evaluation: $\nabla \cdot \mathbf{B} = 0$ means net outward flux through any closed surface is zero, proving isolated single magnetic charges (monopoles) do not exist. This is true.