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Ampere-Maxwell Law

When the circuit with a capacitor is in steady state, 𝐼=0, Therefore 𝜇₀∑𝐼_enclosed=0 ⇒ ∮ 𝐵⋅𝑑𝑙=0
But when the 𝐵-field is moving, 𝜇₀∑𝐼_enclosed≠0 ⇒ ∮ 𝐵⋅𝑑𝑙≠0
Electric flux through this surface: ∫ 𝐸⋅𝑛𝑑𝐴=|𝐸|𝐴=1𝜀₀𝑄𝐴𝐴=𝑄𝜀₀
The time derivative of electric flux is: 𝑑𝑑𝑡∫ 𝐸⋅𝑛𝑑𝐴=𝑑𝑑𝑡𝑄𝜀₀≡1𝜀₀𝐼
So the changing flux acts like a current inside the capacitor. And therefore the line integral of magnetic field is.
∮ 𝐵⋅𝑑𝑙=𝜇₀∑𝐼_enclosed+𝜀₀𝑑𝑑𝑡∫ 𝐸⋅𝑛𝑑𝐴
where the first right term contributes outside the capacitor and the second right term contributes inside the capacitor.

Maxwell's Equations

∮ 𝐸⋅𝑛𝑑𝐴=1𝜀0∑𝑄enclosed; Gauss's Law
∮ 𝐵⋅𝑛𝑑𝐴=0; Gauss's Law (Magnetism)
∮ 𝐸⋅𝑑𝑙=−𝑑𝑑𝑡∫ 𝐵⋅𝑛𝑑𝐴; Faraday's Law
∮ 𝐵⋅𝑑𝑙=𝜇0∑𝐼enclosed+𝜀0𝑑𝑑𝑡∫ 𝐸⋅𝑛𝑑𝐴; Ampere-Maxwell Law Everything there is to know about electricity and magnetism is contained in these four laws plus the force law: 𝐹=𝑞𝐸+𝑞𝑣×𝐵

Differential Form of Gauss's Law

Gauss's Law: ∮ 𝐸⋅𝑛𝑑𝐴=1𝜀₀∑𝑄_enclosed Consider a region of space, enclosed by a box ∆𝑉.
Lim∆𝑉→0∮𝐸⋅𝑛𝑑𝐴∆𝑉=Lim∆𝑉→01𝜀₀∑𝑄_enclosed∆𝑉≡1𝜀₀𝜌
⇒Lim∆𝑉→0∮𝐸⋅𝑛𝑑𝐴∆𝑉=Lim∆𝑉→0(𝐸₂−𝐸₁)∆𝑦∆𝑧∆𝑥∆𝑦∆𝑧=Lim∆𝑥→0(𝐸₂−𝐸₁)∆𝑥≡∂𝐸_𝑥∂𝑥=1𝜀₀𝜌
for a general case where 𝐸 can point in any direction:
∂𝐸_𝑥∂𝑥+∂𝐸_𝑦∂𝑦+∂𝐸_𝑧∂𝑧≡∇⋅𝐸=1𝜀₀𝜌 The parallel derivative of Gauss's Law differential form where ∇≡∂∂𝑥,∂∂𝑦,∂∂𝑧

Differential Form of Ampere's Law

Ampere-Maxwell Law: ∮ 𝐵⋅𝑑𝑙=𝜇₀∑𝐼_enclosed+𝜀₀𝑑𝑑𝑡∫ 𝐸⋅𝑛𝑑𝐴 Consider a region of area, enclosed by a box ∆𝐴 and express 𝐼 in term of 𝐽⋅𝑛∆𝐴.
Lim∆𝐴→0∮𝐵⋅𝑑𝑙∆𝐴=Lim∆𝐴→0𝜇₀𝐽⋅𝑛∆𝐴∆𝐴+Lim∆𝐴→0𝑑𝑑𝑡∫𝐸⋅𝑛𝑑𝐴∆𝐴𝜇₀𝜀₀=Lim∆𝐴→0𝜇₀𝐽_𝑧∆𝐴∆𝐴+Lim∆𝐴→0𝑑𝑑𝑡𝐸_𝑧∆𝐴∆𝐴𝜇₀𝜀₀
⇒Lim∆𝐴→0∮𝐵⋅𝑑𝑙∆𝐴=𝜇₀𝐽_𝑧+𝑑𝐸_𝑧𝑑𝑡𝜇₀𝜀₀
⇒Lim∆𝐴→0∮𝐵⋅𝑑𝑙∆𝐴=Lim∆𝐴→0(𝐵_1,𝑥−𝐵_3,𝑥)∆𝑥+(𝐵_2,𝑦−𝐵_4,𝑦)∆𝑦∆𝑥∆𝑦=𝜇₀𝐽_𝑧+𝑑𝐸_𝑧𝑑𝑡𝜇₀𝜀₀
⇒Lim∆𝐴→0∮𝐵⋅𝑑𝑙∆𝐴=Lim∆𝑦→0(𝐵_1,𝑥−𝐵_3,𝑥)∆𝑦Lim∆𝑥→0(𝐵_2,𝑦−𝐵_4,𝑦)∆𝑥=−∂𝐵_𝑥∂𝑦+∂𝐵_𝑦∂𝑥=𝜇₀𝐽_𝑧+𝑑𝐸_𝑧𝑑𝑡𝜇₀𝜀₀
crossed derivatives: −∂𝐵_𝑥∂𝑦+∂𝐵_𝑦∂𝑥
For a loop in any direction, this can be re-expressed as:
∂𝐵_𝑧∂𝑦−∂𝐵_𝑦∂𝑧𝑥+ ∂𝐵_𝑥∂𝑧−∂𝐵_𝑧∂𝑥𝑦+ ∂𝐵_𝑦∂𝑥−∂𝐵_𝑥∂𝑦𝑧 =∇×𝐵=𝜇₀𝐽+𝜀₀∂𝐸∂𝑡 Ampere's Law differential form

Differential Form of Maxwell's Equations

Divergence for enclosed flux:
∇⋅𝐸=1𝜀0𝜌 for ∮ 𝐸⋅𝑛𝑑𝐴=1𝜀0∑𝑄enclosed; Gauss's Law
∇⋅𝐵=0 for ∮ 𝐵⋅𝑛𝑑𝐴=0; Gauss's Law (Magnetism)
Curl for Circulation:
∇×𝐸=−∂𝐵∂𝑡 for ∮ 𝐸⋅𝑑𝑙=−𝑑𝑑𝑡∫ 𝐵⋅𝑛𝑑𝐴; Faraday's Law
∇×𝐵=𝜇0𝐽+𝜀0∂𝐸∂𝑡 for ∮ 𝐵⋅𝑑𝑙=𝜇0∑𝐼enclosed+𝜀0𝑑𝑑𝑡∫ 𝐸⋅𝑛𝑑𝐴; Ampere-Maxwell Law

Source and Reference

https://www.youtube.com/watch?v=fkfnDopQBYQ&list=PLZ6kagz8q0bvxaUKCe2RRvU_h7wtNNxxi&index=26