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Content

 Maxwell's Equations
 Faraday's Law
  Curly 𝐸 from Moving a Loop of Wire
  Curly 𝐸 from Changing 𝐵
 Lenz's Law
 Source and Reference

Maxwell's Equations

∮ 𝐸⋅𝑛𝑑𝐴=1𝜀₀∑𝑄_enclosed; Gauss's Law

∮ 𝐵⋅𝑛𝑑𝐴=0; Gauss's Law (Magnetism)

 ∮ 𝐸⋅𝑑𝑙=−𝑑𝑑𝑡∫ 𝐵⋅𝑛𝑑𝐴; Faraday's Law

 ∮ 𝐵⋅𝑑𝑙=𝜇₀∑𝐼_enclosed+𝜀₀𝑑𝑑𝑡∫ 𝐸⋅𝑛𝑑𝐴; Ampere-Maxwell Law

In steady state with stationary net charge and constant current, the Maxwell's equations becomes

Inward/Outword Pointing Fields:

        ∮ 𝐸⋅𝑛𝑑𝐴=1𝜀₀∑𝑄_enclosed; Gauss's Law

∮ 𝐵⋅𝑛𝑑𝐴=0; Gauss's Law (Magnetism)

        Curly Fields:

 ∮ 𝐸⋅𝑑𝑙=0

 ∮ 𝐵⋅𝑑𝑙=𝜇₀∑𝐼_enclosed+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

∮ 𝐸⋅𝑑𝑙=−𝑑𝑑𝑡∫ 𝐵⋅𝑛𝑑𝐴

Electromotive Force emf=∮ 𝐸⋅𝑑𝑙

Magnetic Flux Φ≡∫ 𝐵⋅𝑛𝑑𝐴

Therefore Faraday's law can also be defined as emf=−𝑑Φ_mag𝑑𝑡 where − sign means that loops of wire oppose changes in the enclosed or net magnetic flux.

Curly 𝐸 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 𝐸 from Changing 𝐵

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 𝐵_ext and the rate of changes of magnetic field with ∆𝐵_ext=𝐵_ext,final−𝐵_ext,initial. The right hand rule still applies to 𝐸_NC because the − sign is already added to indicate that the loop opposes changes in magnetic flux.
For top-left example: 𝐵_ext is positive and 𝐵_ext is increasing imply −𝑑𝐵_ext𝑑𝑡 is pointing backward, −(+(+)), and by right-hand rule, 𝐸_NC is clockwise.
For top-right example: 𝐵_ext is positive and 𝐵_ext is decreasing imply −𝑑𝐵_ext𝑑𝑡 is pointing forward, −(−(+)), and by right-hand rule, 𝐸_NC is counterclockwise.
For bottom-left example: 𝐵_ext is negative and 𝐵_ext is increasing imply −𝑑𝐵_ext𝑑𝑡 is pointing forward, −(+(−)), and by right-hand rule, 𝐸_NC is counterclockwise.
For bottom-right example: 𝐵_ext is negative and 𝐵_ext is decreasing imply −𝑑𝐵_ext𝑑𝑡 is pointing forward, −(−(−)), and by right-hand rule, 𝐸_NC 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.