Assertion (A): For any closed surface S, \( \sum_{\text{all } \Delta\vec{S}} \vec{B}\cdot\Delta\vec{S}=0 \).
Reason (R): Magnetic monopoles do not exist.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Both (A) and (R) are true and (R) is the correct explanation of (A).
(A) For any closed surface S, \( \sum_{\text{all } \Delta\vec{S}} \vec{B}\cdot\Delta\vec{S}=0 \). — True. This is Gauss’s law for magnetism. The net magnetic flux through any closed surface is zero.
(R) Magnetic monopoles do not exist. — True. An isolated north or south pole has never been found. The simplest magnetic element is a dipole or a current loop.
Why R explains A?
In electrostatics the flux through a closed surface is \( q/\varepsilon_0 \), because field lines begin and end on the charges enclosed.
With no monopoles there are no sources or sinks of \( \vec{B} \). Magnetic field lines form closed loops, so every line that enters a closed surface also leaves it, and the net flux is zero.
NCERT (2026-27):
Magnetism and Gauss’s Law – “… Then, the net flux φB is,
\( \displaystyle \phi_B=\sum_{\text{all}}\Delta\phi_B=\sum_{\text{all}}\mathbf{B}\cdot\Delta\mathbf{S}=0 \)\( \qquad (5.6) \)
where ‘all’ stands for ‘all area elements ΔS’.”
Magnetism and Gauss’s Law – “The difference between the Gauss’s law of magnetism and that for electrostatics is a reflection of the fact that isolated magnetic poles (also called monopoles) are not known to exist. There are no sources or sinks of B; the simplest magnetic element is a dipole or a current loop.”