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โ€ƒMaxwell's Equations
โ€ƒFaraday's Law
โ€ƒโ€ƒCurly ๐ธ from Moving a Loop of Wire
โ€ƒโ€ƒCurly ๐ธ from Changing ๐ต
โ€ƒLenz's Law
โ€ƒSource and Reference

Maxwell's Equations

โˆฎ ๐ธโ‹…๐‘›๐‘‘๐ด=1๐œ€0โˆ‘๐‘„enclosed; Gauss's Law
โˆฎ ๐ตโ‹…๐‘›๐‘‘๐ด=0; Gauss's Law (Magnetism)
โˆฎ ๐ธโ‹…๐‘‘๐‘™=โˆ’๐‘‘๐‘‘๐‘กโˆซ ๐ตโ‹…๐‘›๐‘‘๐ด; Faraday's Law
โˆฎ ๐ตโ‹…๐‘‘๐‘™=๐œ‡0โˆ‘๐ผenclosed+๐œ€0๐‘‘๐‘‘๐‘กโˆซ 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:
โˆฎ ๐ธโ‹…๐‘›๐‘‘๐ด=1๐œ€0โˆ‘๐‘„enclosed; Gauss's Law
โˆฎ ๐ตโ‹…๐‘›๐‘‘๐ด=0; Gauss's Law (Magnetism)
Curly Fields:
โˆฎ ๐ธโ‹…๐‘‘๐‘™=0
โˆฎ ๐ตโ‹…๐‘‘๐‘™=๐œ‡0โˆ‘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.

Faraday's Law

The Faraday's law is โˆฎ ๐ธโ‹…๐‘‘๐‘™=โˆ’๐‘‘๐‘‘๐‘กโˆซ ๐ตโ‹…๐‘›๐‘‘๐ด
Electromotive Force emf=โˆฎ ๐ธโ‹…๐‘‘๐‘™
Magnetic Flux ฮฆโ‰กโˆซ ๐ตโ‹…๐‘›๐‘‘๐ด
Therefore Faraday's law can also be defined as emf=โˆ’๐‘‘ฮฆmag๐‘‘๐‘ก where โˆ’ sign means that loops of wire oppose changes in the enclosed or net magnetic flux.

Curly ๐ธ from Moving a Loop of Wire

image 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

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

Lenz's Law

image The direction of curly electric field depends on the direction of magnetic field ๐ตext and the rate of changes of magnetic field with โˆ†๐ตext=๐ตext,finalโˆ’๐ตext,initial. The right hand rule still applies to ๐ธNC because the โˆ’ 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 โˆ’๐‘‘๐ตext๐‘‘๐‘ก is pointing backward, โˆ’(+(+)), and by right-hand rule, ๐ธNC is clockwise.
For top-right example: ๐ตext is positive and ๐ตext is decreasing imply โˆ’๐‘‘๐ตext๐‘‘๐‘ก is pointing forward, โˆ’(โˆ’(+)), and by right-hand rule, ๐ธNC is counterclockwise.
For bottom-left example: ๐ตext is negative and ๐ตext is increasing imply โˆ’๐‘‘๐ตext๐‘‘๐‘ก is pointing forward, โˆ’(+(โˆ’)), and by right-hand rule, ๐ธNC is counterclockwise.
For bottom-right example: ๐ตext is negative and ๐ตext is decreasing imply โˆ’๐‘‘๐ตext๐‘‘๐‘ก is pointing forward, โˆ’(โˆ’(โˆ’)), 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

https://www.youtube.com/watch?v=BlPctmU1VUg&list=PLZ6kagz8q0bvxaUKCe2RRvU_h7wtNNxxi&index=24

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

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