Electromagnetism

Step 1: Understanding the Concept:
This question requires matching fundamental concepts and equations in electromagnetism. We need to identify the correct mathematical expression for each term in List-I from the options in List-II.
Step 2: Detailed Explanation:

(A) Displacement current ( ): Maxwell's addition to Ampere's law, the displacement current density is defined as . This term accounts for the production of a magnetic field by a time-varying electric field. This matches with (IV).

(B) Poynting vector ( ): The Poynting vector represents the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field. It is defined as . This matches with (III).

(C) Energy stored in electric field ( ): The energy density (energy per unit volume) stored in an electric field is . To find the total energy stored in a volume , we integrate this density over the volume: . This matches with (I).

(D) Gauss's Law: This is one of the four fundamental Maxwell's equations. It relates the divergence of the electric field to the electric charge density . In its differential form, it is . This matches with (II).
Step 3: Final Answer:
Based on the analysis, the correct pairings are:
(A) matches with (IV)
(B) matches with (III)
(C) matches with (I)
(D) matches with (II)
This corresponds to option (B).

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:

: This is Gauss's law for magnetism. It states that the divergence of the magnetic field 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, is the mathematical statement of the non-existence of magnetic monopoles.
: This is Gauss's law for electricity (where in vacuum and 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.
: This is a specific case of Gauss's law for electricity in a region with no charge.
: 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 directly implies that there are no magnetic monopoles, as the net magnetic flux out of any closed surface is always zero.