Step 1: Understanding the Concept:
Maxwell's equations summarize the fundamental laws of electromagnetism. The non-existence of magnetic monopoles implies that magnetic poles always occur in North-South pairs.
Step 2: Key Formula or Approach:
The approach is to evaluate the physical meaning behind the divergence of a vector field. Divergence ($\nabla \cdot$) measures the net source or sink of field lines in a given volume.
Step 3: Detailed Explanation:
Let's analyze the options:
(A) $\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$ is Gauss's Law for electricity, showing electric charges act as sources/sinks for electric fields.
(C) $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$ is Faraday's Law of Induction.
(D) $\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$ is the Amp\`ere-Maxwell Law.
(B) $\nabla \cdot \vec{B} = 0$ is Gauss's Law for magnetism. It states that the divergence of the magnetic field over any volume is exactly zero. This means there are no net magnetic sources or sinks. Consequently, magnetic field lines must form continuous, closed loops, proving that an isolated magnetic monopole cannot exist.
Step 4: Final Answer:
The correct option is (B).