Surface integral is zero for

Question

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Option C: Magnetic flux density

Explanation & Concepts:
The surface integral of magnetic flux density $\mathbf{B}$ over any closed surface is always zero, a property known as Gauss's Law for Magnetism. This physical fact represents the non-existence of magnetic monopoles in nature.

Working Principle: According to Maxwell's second equation in integral form, the net magnetic flux through a closed surface $\oint_S \mathbf{B} \cdot d\mathbf{S} = 0$. This implies that magnetic field lines form continuous closed loops and every 'source' (north pole) is always accompanied by a 'sink' (south pole).

Key Formulas: $\oint_S \mathbf{B} \cdot d\mathbf{S} = 0$ — Gauss's Law for Magnetism (Integral form); $
abla \cdot \mathbf{B} = 0$ — Gauss's Law for Magnetism (Point form)

Key Points: Magnetic field lines are solenoidal in nature, meaning they have no starting or ending points.; The net flux out of a closed surface for a magnetic field is always zero because there are no isolated magnetic charges.; Unlike the electric field (where $\oint_S \mathbf{D} \cdot d\mathbf{S} = Q_{enc}$), there is no 'magnetic charge' enclosed.; This law implies that the magnetic field is divergence-free, i.e., $
abla \cdot \mathbf{B} = 0$.

Advantages: Ensures consistency with the non-existence of magnetic monopoles.; Simplifies the analysis of magnetic circuits using the concept of closed loops.

Applications: Analysis of magnetic flux distribution in motors and transformers.; Verification of field continuity in electromagnetic design.

Additional Info: Electric flux density (Option B) follows Gauss's Law for electrostatics: $\oint_S \mathbf{D} \cdot d\mathbf{S} = Q_{enclosed}$.. Electric field intensity (Option A) is generally not zero over a closed surface if charges are present.. Magnetic field intensity $\mathbf{H}$ (Option D) is related to current via Ampere's Law: $\oint \mathbf{H} \cdot d\mathbf{l} = I$.

Answer: C is correct — The net magnetic flux through any closed surface is always zero because magnetic monopoles do not exist.

Exam Tip: Always remember: Electric flux is associated with charges (divergence), while magnetic flux is associated with current loops (curl, no divergence).

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Option A: Electric field intensity

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Option B: Electric flux density

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Option D: Magnetic field intensity

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