Electromagnetic Theory Part Ⅰ

Basic Laws of Electromagnetic Theory

A charge affects and is affected by other charges around it, and can be thought of as exciting some substance in space through which it interacts, which we call a field. Then we have: $\vec{F_{E}} = q \vec{E}$ , and similarly the magnetic field can be defined: $\vec{F_{B}} = q \vec{v} \times \vec{B}$ .

Faraday's Introduction Law

Obviously a stationary conductor with a stable magnetic field will not generate an electromotive force, otherwise a conductor placed in a magnetic field would spontaneously combust. In addition, we require the conductor to be a closed loop. The magnitude of the electric potential emf is related to the magnetic flux $Φ$ of the non-closed surface enclosed by the conductor and we have the following relation:

\begin{aligned} Φ_{M} = B A \cos θ \\ e m f = - \frac{d Φ}{d t} \end{aligned}

According to the concept of electromotive force, from the perspective of energy, to analyze the charge and electric field, we have:

e m f = - \oint_{C} E d l = - \frac{d Φ}{d t} = - \frac{d}{d t} \iint_{S} \vec{B} d \vec{S}

Obviously, here too, the conductor is required to be a closed loop, and further, we want to examine the relationship between the magnetic field and the electric field itself, so we often bring the conductor to rest, which is quite common, such as placing a coil in a varying magnetic field, and then we have:

- \oint_{C} E d l = \iint_{S} \frac{\partial \vec{B}}{\partial t} d \vec{S}

This suggests that a changing magnetic field over time is accompanied by an electric field

Gauss's Law-Electric

Earlier we discussed the interaction between electric and magnetic fields, next we discuss how charges excite electric fields in space.
It is possible to hypothesize that the electric field is flowing and that the source is charge, borrowing here the idea of equipotential surfaces, and that a flux will be generated by this flowing material as it passes through some surface of equal electric field strength. Experiments have shown that this flow is proportional to the amount of charge contained within this surface:

∯_{A} \vec{E} d \vec{S} = \frac{1}{ϵ_{0}} Σ q = \frac{1}{ϵ_{0}} ∭_{V} ρ d V

Where $ϵ$ is the capacitance, which characterizes the degree of penetration of the material by the electric field in which it is placed, the role of the formula is to make the left and right scales equal, $ϵ_{0}$ is the vacuum capacitance, we can define the dielectric constant $K_{E} = \frac{ϵ_{0}}{ϵ}$ , the magnitude of this value is related to the speed of light in the medium.

Electric Permittivity

Similarly, let's discuss magnetic fields. It is worth noting that the magnetic field lines are always closed, while the electric field lines are not closed, so the electric field flux is not 0 when passing through a closed surface, while the magnetic field flux is 0. We have:

Φ_{M} = ∯_{A} \vec{B} d \vec{S} = 0

Ampère’s Circuital Law

In analogy to the electric field, assuming that there exists a structure like an electron to excite the magnetic field, denote it as $q_{B}$ , and that the magnetic field in the vicinity of an energized wire can be measured as $\frac{μ_{0} i}{2 π r}$ , and let $J$ be the current density, then we have:

\begin{aligned} q_{m} B Σ Δ l = q_{m} B 2 π r = q_{m} \frac{μ_{0} i}{2 π r} 2 π r \\ ⟹ \oint \vec{B} d \vec{l} = \iint_{A} μ_{0} \vec{J} d \vec{S} \end{aligned}

Similarly, $μ_{0}$ is the free space magnetic permeability and we have: $K_{M} = \frac{μ}{μ_{0}}$
However the above equation is not complete because the electric field also excites the magnetic field, which is evident when the capacitor is charging, with $J = 0$ and $B \neq 0$ between the capacitor poles, so we need to correct the above equation. The correction can be made by calculating the magnetic field excited by the electric field between the pole plates alone, or we can speak of the electric field as equivalent to some current density to make Eq. More harmoniously, let the area of the pole plates be $A$ , and we have:

\begin{aligned} {\begin{cases} E = \frac{Q}{ϵ A} \\ \frac{\partial Q}{\partial t} = i \end{cases} ⟹ \frac{\partial \vec{E}}{\partial t} = \frac{i}{A} \\ \vec{J_{D}} = \frac{\partial \vec{E}}{\partial t} \\ ⟹ \oint \vec{B} d \vec{l} = μ \iint_{A} (\vec{J} + \frac{\partial \vec{E}}{\partial t}) d \vec{S} \end{aligned}

This suggests that a time-varying magneto-electric field is accompanied by a magnetic field.

Maxwell's Equation

Consider first the significance of dispersion and spin.

{\begin{cases} \vec{\nabla} = \vec{i} \frac{\partial}{\partial x} + \vec{j} \frac{\partial}{\partial y} + \vec{k} \frac{\partial}{\partial z} \\ \vec{\nabla} E = \frac{\partial E_{x}}{\partial x} \vec{i} + \frac{\partial E_{y}}{\partial y} \vec{j} + \frac{\partial E_{z}}{\partial z} \vec{k} \\ \vec{\nabla} \cdot \vec{E} = \frac{\partial E_{x}}{\partial x} + \frac{\partial E_{y}}{\partial y} + \frac{\partial E_{z}}{\partial z} \\ \vec{\nabla} \times \vec{E} = (\frac{\partial E_{z}}{\partial y} - \frac{\partial E_{y}}{\partial z}) + (\frac{\partial E_{x}}{\partial z} - \frac{\partial E_{z}}{\partial x}) + (\frac{\partial E_{y}}{\partial x} - \frac{\partial E_{x}}{\partial y}) \end{cases}

The gradient is obviously the direction with the largest partial derivative in space, i.e., the direction with the largest rate of change, and the dispersion indicates the degree of dispersion of the field generated by a source, and analysis shows that the dispersion is also the bulk density of the flux. And the magnitude of the spin indicates the degree of rotation of the field, and the direction indicates the direction in which the rotation is maximum under the right-hand rule, and the analysis shows that the spin is the average value of the work done by the field[1,2].

Applying Gaussian formula and stokes we can easily get:

{\begin{cases} - \oint_{C} E d l = \iint_{S} \frac{\partial \vec{B}}{\partial t} d \vec{S} \\ ∯_{A} \vec{E} d \vec{S} = \frac{1}{ϵ_{0}} ∭_{V} ρ d V \\ ∯_{A} \vec{B} d \vec{S} = 0 \\ \oint \vec{B} d \vec{l} = μ \iint_{A} (\vec{J} + \frac{\partial \vec{E}}{\partial t}) d \vec{S} \end{cases} ⟹ {\begin{cases} \nabla \times \vec{E} = - \frac{\partial B}{\partial t} \\ \nabla \cdot \vec{E} = \frac{ρ}{ε_{0}} \\ \nabla \cdot \vec{B} = 0 \\ \nabla \times \vec{B} = (μ \vec{J} + \frac{\partial \vec{E}}{\partial t}) \end{cases}

Electromagnetic Waves

The charge excites the electric field in space, and when the charge is perturbed, the electric field changes and excites the magnetic field, and since the electric field excited by the charge does not vary uniformly with time, the magnetic field excited by it also does not vary uniformly with time, and thus the electric field is excited again, and the above process is repeated again and again, and the conduction is carried out to the farther side, so that the perturbation is carried through the space from this point of the charge, and that the medium does not take part in this conduction, which is very much similar to that of the wave which we have before described as a wave. In fact, we can also derive an electromagnetic wave by means of Maxwell's system of equations, and we can obtain the speed of this wave as $\frac{1}{μ ε_{0}}$ , which corresponds to the speed of light, i.e., light may be an electromagnetic wave.

Transverse Waves

Electromagnetic waves are transverse waves, and let them be plane waves, the electromagnetic field can be represented as: $\vec{E} = {\vec{E}}_{0} e^{i (\vec{k} \cdot \vec{r} - ω t}); \vec{B} = {\vec{B}}_{0} e^{i (\vec{k} \cdot \vec{r} - ω t)};$ . According to the system of Maxwell's equations in vacuum, we have:

\begin{aligned} \vec{\nabla} \cdot f \vec{A} = \vec{A} \nabla f + f \vec{\nabla} \cdot \vec{A} ⟹ \vec{\nabla} \cdot \vec{E} = i \nabla (\vec{k} \vec{r} - ω t) \vec{E_{0}} e^{i (\vec{k} \cdot \vec{r} - ω t)} \vec{E} = i \vec{k} \vec{E_{0}} e^{i (\vec{k} \cdot \vec{r} - ω t)} \\ ⟹ {\begin{cases} \vec{\nabla} \cdot \vec{E} & = i \vec{k} \cdot \vec{E_{0}} e^{i (\vec{k} \cdot \vec{r} - ω t)} = 0 ⟹ \vec{k} \cdot \vec{E_{0}} = 0, \\ \vec{\nabla} \cdot \vec{B} & = i \vec{k} \cdot \vec{B_{0}} e^{i (\vec{k} \cdot \vec{r} - ω t)} = 0 ⟹ \vec{k} \cdot \vec{B_{0}} = 0, \end{cases} \end{aligned}

The direction of the wave vector is perpendicular to the direction of the electric field, i.e. the direction of propagation is perpendicular to the direction of vibration. And further we have:

\begin{aligned} \vec{\nabla} \times f \vec{A} = f (\nabla \times \vec{A}) + (\vec{\nabla} f) \times \vec{A} ⟹ \vec{\nabla} \times \vec{E} = i \vec{k} \times \vec{E_{0}} e^{i (\vec{k} \cdot \vec{r} - ω t)} \\ ⟹ \vec{\nabla} \times \vec{E} = i \vec{k} \times \vec{E_{0}} e^{i (\vec{k} \cdot \vec{r} - ω t)} = - \frac{\partial \vec{B}}{\partial t} = ω \vec{B_{0}} e^{i (\vec{k} \cdot \vec{r} - ω t)} ⟹ i \vec{k} \times \vec{E_{0}} = ω \vec{B_{0}} \\ i \vec{k} \times \vec{B_{0}} = μ ε_{0} ω \vec{E_{0}} \\ ⟹ {\begin{cases} | i \vec{k} \times \vec{B_{0}} | = k E_{0} = ω B_{0} \\ | i \vec{k} \times \vec{B_{0}} | = k B_{0} = μ ε_{0} ω E_{0} \end{cases} ⟹ \frac{E_{0}}{B_{0}} = c \end{aligned}

Also by the cross product we know that the electric field is perpendicular to the magnetic field.

Reference

[1]【微積分-李柏堅-Youtube】https://youtu.be/zKNNGHIju14?si=LqoFFjpJFb4SpW-a
[2]【【nabla算子】与梯度、散度、旋度-哔哩哔哩】 https://b23.tv/XIwgBMl