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 Ampere-Maxwell Law
 Maxwell's Equations
 Differential Form of Gauss's Law
 Differential Form of Ampere's Law
 Differential Form of Maxwell's Equations
 Source and Reference

Ampere-Maxwell Law

image When the circuit with a capacitor is in steady state, 𝐼=0, Therefore 𝜇0𝐼enclosed=0 ⇒ 𝐵⋅𝑑𝑙=0
But when the 𝐵-field is moving, 𝜇0𝐼enclosed≠0 ⇒ 𝐵⋅𝑑𝑙≠0
Electric flux through this surface: 𝐸𝑛𝑑𝐴=|𝐸|𝐴=1𝜀0𝑄𝐴𝐴=𝑄𝜀0
The time derivative of electric flux is: 𝑑𝑑𝑡 𝐸𝑛𝑑𝐴=𝑑𝑑𝑡𝑄𝜀01𝜀0𝐼
So the changing flux acts like a current inside the capacitor. And therefore the line integral of magnetic field is.
𝐵⋅𝑑𝑙=𝜇0𝐼enclosed+𝜀0𝑑𝑑𝑡 𝐸𝑛𝑑𝐴
where the first right term contributes outside the capacitor and the second right term contributes inside the capacitor.

Maxwell's Equations

𝐸𝑛𝑑𝐴=1𝜀0𝑄enclosed; Gauss's Law
𝐵𝑛𝑑𝐴=0; Gauss's Law (Magnetism)
𝐸⋅𝑑𝑙=−𝑑𝑑𝑡 𝐵𝑛𝑑𝐴; Faraday's Law
𝐵⋅𝑑𝑙=𝜇0𝐼enclosed+𝜀0𝑑𝑑𝑡 𝐸𝑛𝑑𝐴; 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: 𝐸𝑛𝑑𝐴=1𝜀0𝑄enclosed image Consider a region of space, enclosed by a box ∆𝑉.
Lim∆𝑉→0𝐸𝑛𝑑𝐴∆𝑉=Lim∆𝑉→01𝜀0∑𝑄enclosed∆𝑉1𝜀0𝜌
Lim∆𝑉→0𝐸𝑛𝑑𝐴∆𝑉=Lim∆𝑉→0(𝐸2−𝐸1)∆𝑦∆𝑧∆𝑥∆𝑦∆𝑧=Lim∆𝑥→0(𝐸2−𝐸1)∆𝑥∂𝐸𝑥∂𝑥=1𝜀0𝜌
for a general case where 𝐸 can point in any direction:
∂𝐸𝑥∂𝑥+∂𝐸𝑦∂𝑦+∂𝐸𝑧∂𝑧𝐸=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=𝜇0𝐼enclosed+𝜀0𝑑𝑑𝑡 𝐸𝑛𝑑𝐴 image Consider a region of area, enclosed by a box ∆𝐴 and express 𝐼 in term of 𝐽𝑛∆𝐴.
Lim∆𝐴→0𝐵⋅𝑑𝑙∆𝐴=Lim∆𝐴→0𝜇0𝐽𝑛∆𝐴∆𝐴+Lim∆𝐴→0𝑑𝑑𝑡𝐸𝑛𝑑𝐴∆𝐴𝜇0𝜀0=Lim∆𝐴→0𝜇0𝐽𝑧∆𝐴∆𝐴+Lim∆𝐴→0𝑑𝑑𝑡𝐸𝑧∆𝐴∆𝐴𝜇0𝜀0
Lim∆𝐴→0𝐵⋅𝑑𝑙∆𝐴=𝜇0𝐽𝑧+𝑑𝐸𝑧𝑑𝑡𝜇0𝜀0
Lim∆𝐴→0𝐵⋅𝑑𝑙∆𝐴=Lim∆𝐴→0(𝐵1,𝑥−𝐵3,𝑥)∆𝑥+(𝐵2,𝑦−𝐵4,𝑦)∆𝑦∆𝑥∆𝑦=𝜇0𝐽𝑧+𝑑𝐸𝑧𝑑𝑡𝜇0𝜀0
Lim∆𝐴→0𝐵⋅𝑑𝑙∆𝐴=Lim∆𝑦→0(𝐵1,𝑥−𝐵3,𝑥)∆𝑦Lim∆𝑥→0(𝐵2,𝑦−𝐵4,𝑦)∆𝑥=−∂𝐵𝑥∂𝑦+∂𝐵𝑦∂𝑥=𝜇0𝐽𝑧+𝑑𝐸𝑧𝑑𝑡𝜇0𝜀0
crossed derivatives: −∂𝐵𝑥∂𝑦+∂𝐵𝑦∂𝑥
For a loop in any direction, this can be re-expressed as:
∂𝐵𝑧∂𝑦∂𝐵𝑦∂𝑧𝑥+ ∂𝐵𝑥∂𝑧∂𝐵𝑧∂𝑥𝑦+ ∂𝐵𝑦∂𝑥∂𝐵𝑥∂𝑦𝑧 =×𝐵=𝜇0𝐽+𝜀0𝐸∂𝑡 Ampere's Law differential form

Differential Form of Maxwell's Equations

Divergence for enclosed flux:
𝐸=1𝜀0𝜌 for 𝐸𝑛𝑑𝐴=1𝜀0𝑄enclosed; Gauss's Law
𝐵=0 for 𝐵𝑛𝑑𝐴=0; Gauss's Law (Magnetism)
Curl for Circulation:
×𝐸=−𝐵∂𝑡 for 𝐸⋅𝑑𝑙=−𝑑𝑑𝑡 𝐵𝑛𝑑𝐴; Faraday's Law
×𝐵=𝜇0𝐽+𝜀0𝐸∂𝑡 for 𝐵⋅𝑑𝑙=𝜇0𝐼enclosed+𝜀0𝑑𝑑𝑡 𝐸𝑛𝑑𝐴; Ampere-Maxwell Law

Source and Reference

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

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ID: 200200502 Last Updated: 2/5/2020 Revision: 0

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