91Ó°ÊÓ

List all four of Maxwell's equations: (i) Gauss's law for electricity: \( \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_{0}} \) (ii) Gauss's law for magnetism: \( \nabla \cdot \vec{B} = 0 \) (iii) Faraday's law of induction: \( \nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} \) (iv) Ampere's law with Maxwell's addition: \( \nabla \times \vec{B} = \mu_{0}(\vec{J} + \epsilon_{0} \frac{\partial \vec{E}}{\partial t}) \)

Interpret the meaning of these equations: (i) Gauss's law for electricity states that the electric field flux through a closed surface is equal to the total charge enclosed divided by the vacuum permittivity. This implies that electric field lines must originate from positive charges and terminate at negative charges. (ii) Gauss's law for magnetism states that the magnetic field flux through a closed surface is equal to zero, meaning there are no isolated magnetic poles (i.e., magnetic field lines form closed loops). (iii) Faraday's law of induction states that a change in the magnetic field through a closed loop will induce an electromotive force (EMF) in the loop, creating an electric field around the loop. (iv) Ampere's law with Maxwell's addition relates the magnetic field around a current-carrying wire to the electric and magnetic fields in the surrounding region.

Analyze each equation to see which one shows that electric field lines do not form closed loops: (i) Gauss's law for electricity: Suggests that electric field lines do not form closed loops, as they must originate and terminate at charges. (ii) Gauss's law for magnetism: This equation describes magnetic field lines rather than electric fields, therefore not applicable. (iii) Faraday's law of induction: This equation shows that the electric field can form closed loops in certain conditions, like in changing magnetic fields. (iv) Ampere's law with Maxwell's addition: This equation describes the magnetic field rather than an electric field, therefore not applicable.

Based on our analysis, the equation that shows that electric field lines do not form closed loops is (i) Gauss's law for electricity. The correct answer is therefore (A) (i) only.