Question

Which of following Maxwell's equation shows non existence of magnetic monopoles?

• A. \( \nabla \cdot \vec{B} = 0 \)
• B. \( \nabla \cdot \vec{D} = \frac{\rho}{\epsilon_0} \)
• C. \( \nabla \cdot \vec{E} = 0 \)
• D. \( \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \)

Hint:

Remember the physical meaning of divergence: it measures the "outflow" of a vector field from a point. Zero divergence for the magnetic field means no point sources or sinks, hence no monopoles. In contrast, the divergence of the electric field is proportional to charge density, indicating that charges are the sources/sinks of the electric field.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Maxwell's equations are a set of fundamental equations that govern electric and magnetic fields.
The question asks which of these equations implies that magnetic monopoles (isolated north or south poles) do not exist.
Step 2: Detailed Explanation:
Let's analyze the given options, which are forms of Maxwell's equations:

\( \nabla \cdot \vec{B} = 0 \): This is Gauss's law for magnetism. It states that the divergence of the magnetic field \( \vec{B} \) is zero. In physical terms, this means there are no "sources" or "sinks" for the magnetic field. Magnetic field lines are always closed loops; they do not start or end at a point. A magnetic monopole, if it existed, would be a source (like a north pole) or a sink (like a south pole) of magnetic field lines, which would result in a non-zero divergence. Therefore, \( \nabla \cdot \vec{B} = 0 \) is the mathematical statement of the non-existence of magnetic monopoles.
\( \nabla \cdot \vec{D} = \rho \): This is Gauss's law for electricity (where \( \vec{D} = \epsilon_0 \vec{E} \) in vacuum and \( \rho \) is the free charge density). It states that electric field lines originate from positive charges and terminate on negative charges. It describes electric monopoles (charges), not magnetic ones.
\( \nabla \cdot \vec{E} = 0 \): This is a specific case of Gauss's law for electricity in a region with no charge.
\( \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \): This is Faraday's law of induction. It describes how a changing magnetic field creates an electric field. It does not relate to the existence of magnetic monopoles.
Step 3: Final Answer:
The equation \( \nabla \cdot \vec{B} = 0 \) directly implies that there are no magnetic monopoles, as the net magnetic flux out of any closed surface is always zero.