Step 1: Concept Clarification:
Maxwell's equations are fundamental laws describing electric and magnetic fields. The query asks which of these equations indicates the absence of magnetic monopoles (independent north or south poles).
Step 2: Equation Analysis:
Evaluating the provided Maxwell's equation forms:
\( abla \cdot \vec{B} = 0 \): This is Gauss's law for magnetism. It signifies that the magnetic field \( \vec{B} \) has zero divergence, meaning no magnetic field sources or sinks exist. Magnetic field lines form closed loops without starting or ending points. A magnetic monopole would act as a source or sink, resulting in non-zero divergence. Thus, \( abla \cdot \vec{B} = 0 \) mathematically proves the non-existence of magnetic monopoles.
\( abla \cdot \vec{D} = \rho \): Gauss's law for electricity (where \( \vec{D} = \epsilon_0 \vec{E} \) in vacuum and \( \rho \) is free charge density). It explains that electric field lines originate from positive charges and terminate on negative charges, thus pertaining to electric monopoles, not magnetic ones.
\( abla \cdot \vec{E} = 0 \): This represents Gauss's law for electricity in a charge-free region.
\( abla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \): Faraday's law of induction, which describes how a changing magnetic field induces an electric field. This law does not address magnetic monopoles.
Step 3: Conclusion:
The equation \( abla \cdot \vec{B} = 0 \) unequivocally states that magnetic monopoles do not exist, as the net magnetic flux through any closed surface is always zero.