Identify each equation in List I

(A) $\oint \vec{B}\cdot \overrightarrow{dl}=\mu_0 i_c+\mu_0\varepsilon_0\frac{d\phi_E}{dt}$ This is the Ampere-Maxwell law.

So, $A \to \text{(IV)}$

Equation (B)

$\oint \vec{E}\cdot \overrightarrow{dl}=\frac{d\phi_B}{dt}$ This is intended to represent Faraday's law of electromagnetic induction.

Standard form is $\oint \vec{E}\cdot d\vec{l}=-\frac{d\phi_B}{dt}$ The negative sign is usually present by Lenz's law, but clearly this option corresponds to Faraday law.

So, $B \to \text{(III)}$

Equation (C)

$\oint \vec{E}\cdot \overrightarrow{dA}=\frac{Q}{\varepsilon_0}$ This is Gauss' law for electricity.

So, $C \to \text{(I)}$

Equation (D)

$\oint \vec{B}\cdot \overrightarrow{dA}=0$ This is Gauss' law for magnetism.

So, $D \to \text{(II)}$

Final matching

$A\!-\!IV,\; B\!-\!III,\; C\!-\!I,\; D\!-\!II$

Compare with options

This matches Option A.

Comparison with stored correct answer

Stored correct answer is A, which agrees with the derived answer.