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`โMaxwell's EquationsโFaraday's LawโโCurly ๐ธ from Moving a Loop of WireโโCurly ๐ธ from Changing ๐ตโLenz's LawโSource and Reference`

## Maxwell's Equations

```โฎ ๐ธโ๐๐๐ด=1๐0โ๐enclosed; Gauss's Law โฎ ๐ตโ๐๐๐ด=0; Gauss's Law (Magnetism) โฎ ๐ธโ๐๐=โ๐๐๐กโซ ๐ตโ๐๐๐ด; Faraday's Law โฎ ๐ตโ๐๐=๐0โ๐ผenclosed+๐0๐๐๐กโซ ๐ธโ๐๐๐ด; Ampere-Maxwell Law ``` In steady state with stationary net charge and constant current, the Maxwell's equations becomes ```Inward/Outword Pointing Fields: โฎ ๐ธโ๐๐๐ด=1๐0โ๐enclosed; Gauss's Law โฎ ๐ตโ๐๐๐ด=0; Gauss's Law (Magnetism) Curly Fields: โฎ ๐ธโ๐๐=0 โฎ ๐ตโ๐๐=๐0โ๐ผenclosed+0; Biot-Savart Law ``` Inward/Outward through closed surface and Curly through boundary of open surface are independent field configurations in constant steady state.

The Faraday's law is ```โฎ ๐ธโ๐๐=โ๐๐๐กโซ ๐ตโ๐๐๐ด Electromotive Force emf=โฎ ๐ธโ๐๐ Magnetic Flux ฮฆโกโซ ๐ตโ๐๐๐ด``` Therefore Faraday's law can also be defined as emf=โ๐ฮฆmag๐๐ก where โ sign means that loops of wire oppose changes in the enclosed or net magnetic flux.

### Curly ๐ธ from Moving a Loop of Wire

In the presence of a perpendicular field, changing the area of a loop of wire causes a motional emf.

### Curly ๐ธ from Changing ๐ต

Moving the loop of wire away from or toward the magnet causes a current to run.

## Lenz's Law

The direction of curly electric field depends on the direction of magnetic field ๐ตext and the rate of changes of magnetic field with โ๐ตext=๐ตext,finalโ๐ตext,initial. The right hand rule still applies to ๐ธNC because the โ sign is already added to indicate that the loop opposes changes in magnetic flux.
For top-left example: ๐ตext is positive and ๐ตext is increasing imply โ๐๐ตext๐๐ก is pointing backward, โ(+(+)), and by right-hand rule, ๐ธNC is clockwise.
For top-right example: ๐ตext is positive and ๐ตext is decreasing imply โ๐๐ตext๐๐ก is pointing forward, โ(โ(+)), and by right-hand rule, ๐ธNC is counterclockwise.
For bottom-left example: ๐ตext is negative and ๐ตext is increasing imply โ๐๐ตext๐๐ก is pointing forward, โ(+(โ)), and by right-hand rule, ๐ธNC is counterclockwise.
For bottom-right example: ๐ตext is negative and ๐ตext is decreasing imply โ๐๐ตext๐๐ก is pointing forward, โ(โ(โ)), and by right-hand rule, ๐ธNC is clockwise. And the lenz's Law: A loop of wire tends to oppose changes of magnetic flux through it. And top-left is easier to remember.

## Source and Reference

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ID: 200102902 Last Updated: 1/29/2020 Revision: 0

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