If magnetic monopoles existed, how would the Gauss's law in magnetism be modified?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \oint_{S} \vec{B}\cdot d\vec{s} = \mu_0 q_m \); where \( q_m \) is monopole magnetic charge enclosed by \( S \).
Gauss's law in magnetism says that the net magnetic flux through any closed surface is zero, because isolated magnetic poles do not exist. If monopoles existed, the flux would equal \( \mu_0 \) times the enclosed magnetic charge, just as the electric flux equals \( q/\varepsilon_0 \):
\( \displaystyle \oint_S \vec{B}\cdot d\vec{s} = \mu_0\,q_m \)
Option (1) \( \oint_{S} \vec{B}\cdot d\vec{s} = \mu_0 I \); where \( I \) is the current enclosed by \( S \). — Incorrect. An enclosed current belongs to Ampere's circuital law, a line integral round a loop.
Option (3) \( \oint_{S} \vec{B}\cdot d\vec{l} = \mu_0 q_m \); where \( q_m \) is monopole magnetic charge enclosed by \( S \). — Incorrect. \( \oint \vec{B}\cdot d\vec{l} \) is a line integral around a loop, not the flux through a closed surface.
Option (4) \( \oint_{S} \vec{B}\cdot d\vec{s} = \displaystyle\frac{q_m}{\mu_0} \); where \( q_m \) is monopole magnetic charge enclosed by \( S \). — Incorrect. The magnetic charge is multiplied by \( \mu_0 \), not divided by it.
NCERT (2026-27):
Magnetism and Gauss’s Law – “… Gauss’s law of magnetism states that the flux of B through any closed surface is always zero
\( \displaystyle \oint_{S} \mathbf{B}\cdot d\mathbf{s} = 0 \)
If monopoles existed, the right hand side would be equal to the monopole (magnetic charge) qm enclosed by S. [Analogous to Gauss’s law of electrostatics,
\( \displaystyle \oint_{S} \mathbf{B}\cdot d\mathbf{s} = \mu_0\,q_m \)
where qm is the (monopole) magnetic charge enclosed by S.]”