1. Introduction: A review on static electric and magnetic fields
Coulomb’s law and static electric field
Current and charge conservation
Biot-Savert’s law and static magnetic field
Incompletion of the static description: Should E/M interactions stay independently or they are somehow related?

2. Coulomb’s law Coulomb’s law： A general expression:
Why does the force/field follow an inverse-square dependence on the distance?
definition of the E-field
with charge e1=Ze positioned in the origin
for continuous charge distribution with a volume density of ρ

3. The inverse-square law: a must of our 3D universe
Total force flux must be conserved in a 3D space and equal to the source (i.e., total “detected” = total “created”)
As the distance (r) increases, the surface area (of a sphere that has the source enclosed) increases at a square rate (4πr2). The force therefore have to drop at a square rate to make the flux a constant, as the source is a constant.
We immediately have Gaussian law:
Coulomb force in space with other dimensions?

4. The conservation (path independent) property
From Coulomb’s law we can derive Gaussian law, but the reverse is not sufficient: what is the missing feature?
The coulomb force is a centric force, and our space is isotropic. The static electric field is then curl-free:
More features? - No
From Helmholtz’s law, a vector is uniquely specified by its divergence and curl, hence Gaussian law plus curl-free description is equivalent to Coulomb’s law

5. Home work
Prove Helmholtz’s theorem:

6. The introduction of potential
Since a gradient field is curl-free, we can then introduce a scalar variable and express the static E-field as the gradient of the new (scalar) variable (potential)
We then have three equivalent descriptions on the static E-field:
(a) Coulomb’s law (known as the direct or explicit description)
(b) Gaussian law + the curl-free condition (known as the constraint or implicit description)
(c) Potential description

7. Multifaceted descriptions of the static E-field
The constraint description
Field is specified in a limited set of spatial points, not specified in the rest area (Note: Not specified doesn’t necessarily mean that the field is zero)
Specification is given in the form of equations – implicit expressions
These equations must be in a differential or integral form, cannot be in an algebraic form (why?)
Why go from the direct (explicit) description to the constraint (implicit) and potential description?

8. Reasons For the constraint description For the potential description
We have to deal with the coupling between the electric and magnetic field later: it is easier to deal with coupling problems if we express source by field, rather than to express field by source
For the potential description
We only need to deal with a scalar field (with 1 variable), rather than a vector field (with 3 variables)
Drawback: we have to deal with a 2nd order differential equation rather than the 1st order differential equation: for any numerical solution approach, a much more dense grid is required to treat the oscillatory type solution associated with the 2nd order DE (as contrasted with the smooth solution to the 1st order DE: charge or discharge curve in an exponential shape)

9. What will happen if the source is moving?
Current definition:
Total charge must be conserved, the net current flow through a closed surface must be equal to the change of the charge density enclosed by the surface. Hence we have the charge conservation law:

10. Biot-Savert’s law
There exists a new force (different from the Coulomb’s force) between two currents, described by Biot-Savert’s law:
Biot-Savert force is similar to Coulomb’s force, only the scalar charges are replaced by the current flow vectors; the magnetic force, again, follows the inverse square law as a function of the interactive distance
definition of the M-field
by following the convention: source appears at r’, the tester for probing the force is positioned at r

11. Biot-Savert’s law
Coulomb’s force is originated by the point sources (and their superposition), whereas Biot-Savert’s force comes from the vortex sources (and their superposition) – a vortex cannot be reduced to a single point, and exists only in space with dimension higher than 2!
Biot-Savert’s force only acts on current flows, has no effect on static charges, hence is an independent force (cannot be included by Coulomb’s force)
The E-field bears the nature of a polar vector, whereas the M-field bears the nature of an axial vector.

12. From explicit to implicit
Feature 1: closed vector flow
Magnetic field is continuous in 3D space → divergence free
Vector potential can be introduced, and we have Gaussian law for magnetic field:

13. From explicit to implicit
Closed flow =
Divergence free =
No single magnetic pole =
Zero flux on any enclosed surface in 3D space
but what about this guy?
A Klein bottle: enclosed surface without partitioning the 3D space into 2 parts

14. From explicit to implicit
Feature 2: non-centric field anymore, otherwise, divergence free plus curl free makes no magnetic field anywhere (following Helmholtz’s theorem)
The M-field must have non-zero curl - we can derive Ampere’s law to see this explicitly

15. A contrast of static E- and M- fields
Electric field interacts between two static charges, reflects a pure “adiabatic” effect
Magnetic field interacts between two moving charges, reflects a pure “derivative” effect
Electric + Magnetic field provides a complete description
Static E/M fields form a complementary pair: complete and orthogonal

16. A contrast of static E- and M- fields
Static charge distribution creates a curl free, divergence driven field; the field can be detected by another charge, hence it is called “electric” field
Constant current flow not only creates an electric field, it also creates a divergence free, curl driven field; the field can be detected by another current, hence it is called “magnetic” field
The static electric and magnetic field have no coupling in between, as if they are two different phenomena

17. A summary of static E/M fields
The static E-field
The static M-field
Direct (explicit)
description
Constraint (implicit)
description
Potential description
Relations between
field and potential

18. Home work The static E-field The static M-field Differential form
Prove!
Integral form

19. Incompletion of the static description
However, there is no absolute inertia system – the status, moving or steady, depends on the system where the observer stays, hence a static charge and a constant current can be exchanged in different reference systems (where the observer stays)
Therefore, the static electric field comes from a static charge and the static magnetic field comes from a constant current can be exchanged!
How can we have a consistent description then?

20. The approach for a consistent description
Step 1: introduce the time dependence, find governing equations for the time-varying fields
We will then obtain the Maxwell equations. However, the aforementioned question is only solved in an implicit way.
Step 2: unify electric and magnetic fields as a tensor in a unified spatial - temporal 4D space
We will obtain a unified theory without any inconsistency: the special theory of relativity. Under this picture, the aforementioned problem naturally disappears as the electric and magnetic fields are just different components of a same physical quantity (the EM tensor)!

21. A consistent description
As a special case of the STR, under the non-relativistic condition (v<<c), we have:
Either in the original system or in a moving system (at a speed of v relative to the original system), we will see a consistent solution as E/B and E’/B’ in the reference and moving system, respectively, with E/B and E’/B’ transformed by the above equation