1. Time-varying fields and coupling between E/M fields Maxwell’s equations

2. Faraday’s law
Accelerated electron creates time-varying current, time-varying current creates time-varying magnetic field, time-varying magnetic field will create electric field vortex
Explanation – sourceless and isolated natural system tend to eliminate any disturbance, with time evolution towards its eigen state: if a time-varying magnetic field is enforced without any other source involved, an electric field vortex will have to be induced, to establish yet another magnetic field with an opposite change towards a cancellation of the total field, hence Faraday’s law holds
electromotive force or emf

3. A second way to generate E field
Accelerated charge causes a time-varying charge distribution, leading to a time-varying current; from Ampere’s law, time-varying magnetic field exists; from Faraday’s law, electric vortex exists
Therefore, not only electric field can be generated in its divergence form by the static charge distribution, it can also be generated in its curl form by the “temporary” charge distribution
The electric field is brought in by both its divergence and curl, following the Helmholtz theorem, the electric field is complete
Hence, time-varied magnetic field excites electric field, electric and magnetic fields become coupled; however, such coupling is still unidirectional

4. A map showing all relations
Static
charge
Moving charge, static charge
distribution (constant current)
Time-varying charge
distribution (time-varying
current, temporary charge)
Static Electric
Field
Div.
Static Magnetic
Curl
Time-varying
magnetic field
electric field
Charge conservation law
Gauss’s law
Ampere’s law
Faraday’s law
(Derived from Coulomb’s law)
(Derived from Biot-Savert’s law)

5. Displacement current
Contradiction found between Ampere’s law and the charge conservation law
Maxwell mended Ampere’s law, solved the problem
Gaussian law is called

6. Mutual coupling between dynamic E/M fields
Significance – similar to a current, a time-varying electric field can generate magnetic field
Hence the time-varying rate of the electric displacement vector is equivalent to a current, named as the displacement current; the conventional current caused by the moving charge is then called the conduction current to make a difference
Not only time-varying magnetic field, equivalent to temporary charges, can excite electric field, time-varying electric field, equivalent to a “temporary” current (i.e., the displacement current), can also excite magnetic field
Finally, electric and magnetic fields are fully coupled through mutual excitation

7. Significance of E/M mutual coupling
Therefore
1. accelerated charge creates time-varying magnetic field in its neighborhood (through the time-varying conduction current and following Ampere’s law);
2. time-varying magnetic field excites electric field in its neighborhood (through Faraday’s law)
3. time-varying electric field excites magnetic field in its neighborhood (through Ampere’s law)
Step 2 and 3 form a sustainable loop, and make electric-magnetic fields propagation! Such an electric-magnetic non-local oscillation is named as the electromagnetic wave

8. Maxwell’s equations in vacuum: differential form
Faraday’s law
Modified Ampere’s law (by Maxwell),
derived from Biot-Savert’s law
Gaussian law, derived from Coulomb’s law
Gaussian law, derived from Biot-Savert’s law
8 equations with one in redundant, e.g., the 4th equation is embedded in the 1st equation

9. Maxwell’s equations in vacuum: integral form
Faraday’s law
Modified Ampere’s law (by Maxwell),
derived from Biot-Savert’s law
Gaussian law, derived from Coulomb’s law
Gaussian law, derived from Biot-Savert’s law
The differential form is more general.
The integral form is usually ill-posed, can only be used when the system has a high symmetry.
Take it as a home work to prove the integral form from the differential form.