Editing Electrical length (section)

== Regimes of electromagnetics ==
The field of [[electromagnetics]] is the study of [[electric field]]s, [[magnetic field]]s, [[electric charge]], [[electric current]]s and [[electromagnetic wave]]s.  Classic electromagnetism is  based on the solution of [[Maxwell's equations]].  These equations are mathematically difficult to solve in all generality, so approximate methods have been developed that apply to situations in which the electrical length of the apparatus is very short (<math>G \ll 1</math>) or very long (<math>G \gg 1</math>).  Electromagnetics is divided into three regimes or [[fields of study]] depending on the electrical length of the apparatus, that is the physical length <math>l</math> of the apparatus compared to the wavelength <math>\lambda = c/f</math> of the waves:<ref name="Schmitt" />{{rp|p.21}}<ref name="Azadeh">{{cite book
 | last1  = Azadeh
 | first1 = Mohammad 
 | title  = Fiber Optics Engineering
 | publisher = Springer Science and Business Media
 | date   = 2009
 | pages  = 11
 | url    = https://books.google.com/books?id=8yV-Gwa2ByMC&dq=%22circuit+theory%22+microwave++optics+electromagnetics+wavelength&pg=PA11
 | doi    = 
 | id     = 
 | isbn   =  9781441903044
 }}</ref><ref name="Pozar">{{cite book
 | last1  = Pozar
 | first1 = David M. 
 | title  = Microwave Engineering, 4th Ed.
 | publisher = Wiley Global Education
 | date   = 2011
 | location = 
 | pages  = 1–2
 | url    = https://books.google.com/books?id=JegbAAAAQBAJ&dq=%22circuit+theory%22+microwave+optics&pg=PA1
 | doi    = 
 | id     = 
 | isbn   = 9781118213636
 }}</ref><ref name="Karmel">{{cite book
 | last1  = Karmel
 | first1 = Paul R.
 | last2  = Colef
 | first2 = Gabriel D.
 | last3  = Camisa
 | first3 = Raymond L.
 | title  = Introduction to Electromagnetic and Microwave Engineering
 | publisher = John Wiley and Sons
 | date   = 1998
 | location = 
 | pages  = 1–2
 | url    = https://books.google.com/books?id=iruLnH941OEC&dq=%22circuit+theory%22+microwave++optics&pg=PA1
 | doi    = 
 | id     = 
 | isbn   = 9780471177814
 }}</ref>  Completely different apparatus is used to conduct and process electromagnetic waves in these different wavelength ranges
*<math>\lambda \gg l</math> ''[[Circuit theory]]'': When the wavelength of the electrical oscillations is much larger than the physical size of the circuit (<math> G \ll 1</math>), say <math>\lambda > 50l</math>,<ref name="Clark">{{cite book
 |last1        = Clark
 |first1       = Alan Robert
 |last2        = Fourie
 |first2       = Andre P. C.
 |title        = Antennas in Practice
 |publisher    = Poynting Innovations
 |date         = 2001
 |pages        = 3
 |language     = 
 |url          = https://poynting.tech/wp-content/uploads/downloads/reference_materials/Antennas-in-Practice.pdf
 |doi          = 
 |id           = 
 |isbn         = 0620276193
 |archive-date = 2023-01-02
 |access-date  = 2023-01-02
 |archive-url  = https://web.archive.org/web/20230102172354/https://poynting.tech/wp-content/uploads/downloads/reference_materials/Antennas-in-Practice.pdf
 |url-status   = dead
}}</ref> the action occurs in the [[Near and far field|near field]].   The [[phase (waves)|phase]] of the oscillations and therefore the current and voltage can be approximated as constant along the length of connecting wires.  Also little energy is radiated in the form of [[electromagnetic wave]]s, the power radiated by a conductor as an antenna is proportional to the electrical length squared <math>(l/\lambda)^2 = G^2</math>.   So the electrical energy remains in the wires and components as [[quasistatic approximation|quasistatic]] near-field [[electric field|electric]] and [[magnetic field]]s.  Therefore, the approximation of the [[lumped element model]] can be used, and electric currents  oscillating at these frequencies can be processed by [[electric circuit]]s consisting of lumped [[circuit element]]s such as resistors, capacitors, inductors, transformers, transistors, and [[integrated circuit]]s linked by ordinary wires.  Mathematically Maxwell's equations reduce to [[circuit theory]] ([[Kirchhoff's circuit laws]]).   
*<math>\lambda \approx l</math>, ''[[Distributed-element model]] ([[microwave]] theory)'': When the wavelength of the waves is of the same order of magnitude as the size of the equipment (<math> G \approx 1</math>),  as it is in the [[microwave]] part of the spectrum, full solutions of Maxwell's equations must be used.  At these frequencies, wires are replaced by [[transmission line]]s and [[waveguide]] and lumped elements are replaced by [[resonant stub]]s, irises, and [[microwave cavity|cavity resonators]].  Often only a single [[mode (electromagnetism)|mode]] (wave pattern) is propagating through the apparatus, which simplifies the mathematics.  A modification of circuit theory called the [[distributed-element model]] can often be used, in which extended objects are regarded as electrical circuits with capacitance, inductance and resistance distributed along their length.   A graphical aid called the [[Smith chart]] is often used to analyze transmission lines. 
*<math>\lambda \ll l</math>, ''[[Optics]]'': When the wavelength of the electromagnetic wave is much smaller than the physical size of the equipment that manipulates it (<math> G \gg 1</math>), say <math>\lambda < l/50</math>, most of the path of the waves is in the [[Near and far field|far field]].  In the far field, the electric and magnetic fields cannot be separated but propagate together as an electromagnetic wave.  Unlike in the case of microwaves, unless [[coherent light]] sources like lasers are used, the number of [[mode (electromagnetism)|mode]]s propagating is usually large. Since little of the energy is stored in the [[quasistatic approximation|quasistatic]] (induction) electric or magnetic fields at the surface boundaries between media (called [[evanescent field]]s in optics), the concepts of voltage, current, capacitance, and inductance have little meaning and are not used, and the medium is characterized by its [[index of refraction]] <math>\nu = c/v_\text{p} = \sqrt{\epsilon_\text{r}\mu_\text{r}}</math>, absorption, [[permittivity]] <math>\epsilon</math>, [[Magnetic permeability|permeability]] <math>\mu</math>, and [[Dispersion (optics)|dispersion]].  At these frequencies electromagnetic waves are manipulated by optical elements such as [[lens]]es, mirrors, [[Prism (optics)|prism]]s, [[optical filter]]s and [[diffraction grating]]s.   Maxwell's equations can be approximated by the equations of [[geometrical optics]] or [[physical optics]].
Historically, electric circuit theory and optics developed as separate branches of physics until at the end of the 19th century [[James Clerk Maxwell]]'s electromagnetic theory and [[Heinrich Hertz]]'s discovery that light was electromagnetic waves unified these fields as branches of electromagnetism.