Classical electromagnetism
Classical electromagnetism (or classical electrodynamics) is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model. The theory provides an excellent description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics.
Fundamental physical aspects of classical electrodynamics are presented in many texts, such as those by Feynman, Leighton and Sands, Panofsky and Phillips, and Jackson.
History
The physical phenomena that electromagnetism describes have been studied as separate fields since antiquity. For example, there were many advances in the field of optics centuries before light was understood to be an electromagnetic wave. However, the theory of electromagnetism, as it is currently understood, emerged as a unified field over the course of the 19th century, most prominently in a set of equations systemized by James Clerk Maxwell. For a detailed historical account, consult Pauli, Whittaker, and Pais.
Lorentz force
The electromagnetic field exerts the following force (often called the Lorentz force) on charged particles:
F = qE + qv * B
where all boldfaced quantities are vectors: F is the force that a charge q experiences, E is the electric field at the location of the charge, v is the velocity of the charge, B is the magnetic field at the location of the charge.
The above equation illustrates that the Lorentz force is the sum of two vectors. One is the cross product of the velocity and magnetic field vectors. Based on the properties of the cross product, this produces a vector that is perpendicular to both the velocity and magnetic field vectors. The other vector is in the same direction as the electric field. The sum of these two vectors is the Lorentz force.
Therefore, in the absence of a magnetic field, the force is in the direction of the electric field, and the magnitude of the force is dependent on the value of the charge and the intensity of the electric field. In the absence of an electric field, the force is perpendicular to the velocity of the particle and the direction of the magnetic field. If both electric and magnetic fields are present, the Lorentz force is the sum of both of these vectors.
The electric field E
The electric field E is defined such that, on a stationary charge:
F = q0E
where q0 is what is known as a test charge. The size of the charge doesn't really matter, as long as it is small enough not to influence the electric field by its mere presence. What is plain from this definition, though, is that the unit of E is N/C (newtons per coulomb). This unit is equal to V/m (volts per meter); see below.
In electrostatics, where charges are not moving, around a distribution of point charges, the forces determined from Coulomb's law may be summed. The result after dividing by q0 is:
1 n qi ( r - ri )
E(r) = ______ Σ _________
4πε0 i=1 │r - ri │^3
where n is the number of charges, qi is the amount of charge associated with the ith charge, ri is the position of the ith charge, r is the position where the electric field is being determined, and ε0 is the electric constant.
If the field is instead produced by a continuous distribution of charge, the summation becomes an integral:
1 ρ(r')(r-r')
E(r) = ______ ∫ __________ d^3r'
4πε0 |r-r'|^3
where ρ(r') is the charge density and r-r' is the vector that points from the volume element d^3r' to the point in space where E is being determined.
Both of the above equations are cumbersome, especially if one wants to determine E as a function of position. A scalar function called the electric potential can help. Electric potential, also called voltage (the units for which are the volt), is defined by the line integral
φ(r) = - ∫_c E * dl
where φ(r) is the electric potential, and C is the path over which the integral is being taken.
Unfortunately, this definition has a caveat. From Maxwell's equations, it is clear that ∇ × E is not always zero, and hence the scalar potential alone is insufficient to define the electric field exactly. As a result, one must add a correction factor, which is generally done by subtracting the time derivative of the A vector potential described below. Whenever the charges are quasistatic, however, this condition will be essentially met.
From the definition of charge, one can easily show that the electric potential of a point charge as a function of position is:
1 n qi
φ(r) = _____ Σ _____
4πε0 i=1 | r - ri |
where q is the point charge's charge, r is the position at which the potential is being determined, and ri is the position of each point charge. The potential for a continuous distribution of charge is:
1 ρ(r')
φ(r) = _____ ∫ _____ d^3 r'
4πε0 |r-r'|
where ρ(r) is the charge density, and r - ri is the distance from the volume element d^3 r' to point in space where φ is being determined.
The scalar φ will add to other potentials as a scalar. This makes it relatively easy to break complex problems down in to simple parts and add their potentials. Taking the definition of φ backwards, we see that the electric field is just the negative gradient (the del operator) of the potential. Or:
E(r) = - ∇ φ(r)
From this formula it is clear that E can be expressed in V/m (volts per meter).
Electromagnetic waves
A changing electromagnetic field propagates away from its origin in the form of a wave. These waves travel in vacuum at the speed of light and exist in a wide spectrum of wavelengths. Examples of the dynamic fields of electromagnetic radiation (in order of increasing frequency): radio waves, microwaves, light (infrared, visible light and ultraviolet), x-rays and gamma rays. In the field of particle physics this electromagnetic radiation is the manifestation of the electromagnetic interaction between charged particles.
General field equations
As simple and satisfying as Coulomb's equation may be, it is not entirely correct in the context of classical electromagnetism. Problems arise because changes in charge distributions require a non-zero amount of time to be "felt" elsewhere (required by special relativity).
For the fields of general charge distributions, the retarded potentials can be computed and differentiated accordingly to yield Jefimenko's Equations.
Retarded potentials can also be derived for point charges, and the equations are known as the Liénard–Wiechert potentials. The scalar potential is:
1 q
φ = ______ ______________________________________________
4πε0 v_q ( t_ret )
| r - r_q ( t_ret ) | - ___________ * ( r - r_q ( t_ret ))
c
where q is the point charge's charge and r is the position. rq and vq are the position and velocity of the charge, respectively, as a function of retarded time. The vector potential is similar:
μ0 q v_q ( t_ret )
A = ____ ______________________________________________
4π v_q ( t_ret )
| r - r_q ( t_ret ) | - ___________ * ( r - r_q ( t_ret ))
c
These can then be differentiated accordingly to obtain the complete field equations for a moving point particle.
Models
Branches of classical electromagnetism such as optics, electrical and electronic engineering consist of a collection of relevant mathematical models of different degrees of simplification and idealization to enhance the understanding of specific electrodynamics phenomena, cf. An electrodynamics phenomenon is determined by the particular fields, specific densities of electric charges and currents, and the particular transmission medium. Since there are infinitely many of them, in modeling there is a need for some typical, representative
(a) electrical charges and currents, e.g. moving pointlike charges and electric and magnetic dipoles, electric currents in a conductor etc.;
(b) electromagnetic fields, e.g. voltages, the Liénard–Wiechert potentials, the monochromatic plane waves, optical rays; radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, gamma rays etc.;
(c) transmission media, e.g. electronic components, antennas, electromagnetic waveguides, flat mirrors, mirrors with curved surfaces convex lenses, concave lenses; resistors, inductors, capacitors, switches; wires, electric and optical cables, transmission lines, integrated circuits etc.;
all of which have only few variable characteristics.
electromagnetism 電磁気
electrodynamics 電気力学
branch 枝
theoretical 理論的な
studies 研究
interaction 相互作用
extension 延長
whenever 何時でも
relevant 関連した
negligible 取るに足らない néglɪdʒəbl
aspect 側面
antiquity 古さ
advance 進歩
optics 光学
emerge 出てくる
unify 一つにする
prominently 顕著に
consult 意見を聞く
exert 働かせる
boldface ボールド体
property 財産
perpendicular 垂直
therefore したがって
absence 不在
dependent 依存
intensity 強度
present 現在
stationary 定常の
influence 影響
mere ほんの
definition 定義
though しかし
below 以下に
electrostatics 静電気学
distribution 分布
determine 決心する
sum 合計
divide 分ける
associate 関係づける
constant 不変の
instead その代わりとして
continuous 連続的な
summation 合計
integral 積分の
density 密集
cumbersome 扱いにくい
especially 特に
function 機能 関数
voltage 電圧
path 通り道
unfortunately 不幸にも
caveat 警告
hence このゆえに
insufficient 不十分な
exactly 正確に
correction 訂正
subtract 引く
derivative 派生的な
describe 記述する
whenever いつでも
quasistatic 準静的
essentially 本質的に
met 会う meet
relatively 相対的に
potential 電位の
backwards 後方の
gradient 傾き
operator 演算子
formula 式
express 表わす
propagate 伝播する
spectrum 波帯
wavelength 波長
dynamic 動的な
radiation 輻射
frequency 周波数
infrared 赤外線の
manifestation 表明
interaction 相互作用
entirely 完全に
correct 正確な
context 文脈
arise 起こる
felt 羅紗
elsewhere どこかよそに
retarde 遅らせる
compute 計算する
differentiate 識別する
accordingly それに応じて
derive 引き出す
respectively それぞれ
similar 同様の
obtain 得る
consist 成る
relevant 関連のある
degree 程度
simplification 平易化
idealization 理想化する
enhance 高める
specific 明確な
transmission 伝達
infinitely 無限に
typical 典型的な
representative 代表する
pointlike 点状 concave 凹面 convex 凸状の
dipole 双極子 resistor 抵抗器
conductor 伝導体 inductor 誘導子
monochromatic 単色の capacitor 蓄電器
media 媒体 integrate 統合する
component 構成要素 variable 変数の
waveguide 導波管 characteristic 特質ある