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Match List-I with List-II (the symbols carry their usual meaning).

List-I (A) ∮E⃗⋅dA⃗=Qε0\oint \vec{E} \cdot d\vec{A} = \frac{Q}{\varepsilon_0} (B) ∮B⃗⋅dA⃗=0\oint \vec{B} \cdot d\vec{A} = 0 (C) ∮E⃗⋅dl⃗=−dϕdt\oint \vec{E} \cdot d\vec{l} = -\frac{d\phi}{dt} (D) ∮B⃗⋅dl⃗=μ0ic+μ0ε0dϕEdt\oint \vec{B} \cdot d\vec{l} = \mu_0 i_c + \mu_0 \varepsilon_0 \frac{d\phi_E}{dt}

List-II (I) Ampere-Maxwell's law (II) Faraday's law (III) Gauss's law of electrostatics (IV) Gauss's law of magnetism

A

A-III, B-IV, C-II, D-I

B

A-IV, B-III, C-II, D-I

C

A-III, B-II, C-IV, D-I

D

A-IV, B-I, C-III, D-II

Step-by-Step Solution

  1. Equation A: ∮E⃗⋅dA⃗=Q/ε0\oint \vec{E} \cdot d\vec{A} = Q/\varepsilon_0 represents the total electric flux through a closed surface being proportional to the enclosed charge. This is Gauss's Law of Electrostatics (III).
  2. Equation B: ∮B⃗⋅dA⃗=0\oint \vec{B} \cdot d\vec{A} = 0 indicates that the net magnetic flux through any closed surface is zero, implying magnetic monopoles do not exist. This is Gauss's Law of Magnetism (IV).
  3. Equation C: ∮E⃗⋅dl⃗=−dϕ/dt\oint \vec{E} \cdot d\vec{l} = -d\phi/dt describes how a time-varying magnetic flux induces an electric field. This is Faraday's Law of Induction (II).
  4. Equation D: ∮B⃗⋅dl⃗=μ0ic+μ0ε0dϕEdt\oint \vec{B} \cdot d\vec{l} = \mu_0 i_c + \mu_0 \varepsilon_0 \frac{d\phi_E}{dt} is the generalized form of Ampere's Circuital Law, including the displacement current term introduced by Maxwell. This is the Ampere-Maxwell Law (I). Therefore, the correct matching sequence is A-III, B-IV, C-II, D-I.
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