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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

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

```∮ 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: 𝐸𝑛𝑑𝐴=1𝜀0𝑄enclosed 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 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 ∮ 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

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

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