This question asks to associate the provided Maxwell's equations with their corresponding laws. Below is a breakdown of each equation and its classification:
\(\oint \vec{B} \cdot d\vec{l} = \mu_0 i_c + \mu_0 \epsilon_0 \frac{d\phi_E}{dt}\) corresponds to the Ampere-Maxwell Law. This law integrates Ampere's circuital law with Maxwell's displacement current, accounting for time-varying electric fields, and describes the generation of magnetic fields by electric currents and changing electric fields. \(\rightarrow\)IV. Ampere-Maxwell law.
\(\oint \vec{E} \cdot d\vec{l} = -\frac{d\phi_B}{dt}\) represents Faraday's Law of Electromagnetic Induction, stating that the line integral of the electric field around a closed loop equals the negative rate of change of magnetic flux, explaining the induction of electric fields by changing magnetic fields. \(\rightarrow\)III. Faraday law.
\(\oint \vec{E} \cdot d\vec{A} = \frac{Q}{\epsilon_0}\) is Gauss's Law for electricity, which posits that the electric flux through a closed surface is proportional to the enclosed electric charge. \(\rightarrow\)I. Gauss' law for electricity.
\(\oint \vec{B} \cdot d\vec{A} = 0\) signifies Gauss's Law for magnetism, indicating zero net magnetic flux through any closed surface, thereby confirming the absence of magnetic monopoles. \(\rightarrow\)II. Gauss' law for magnetism.
Matching List I with List II yields the following pairings:
A-IV: Equation A represents the Ampere-Maxwell Law.
B-III: Equation B corresponds to Faraday's Law.
C-I: Equation C is representative of Gauss's Law for electricity.
D-II: Equation D fits with Gauss's Law for magnetism.
The correct answer is therefore A-IV, B-III, C-I, D-II.
