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{{cite web |title=What is "Stray Voltage"? |date=August 10, 2015 |publisher=Utility Technology Engineers-Consultants (UTEC) |url=https://www.utilitytec.com/uploads/1/3/0/9/130926103/2015_stray_voltage_analysis.pdf |access-date=December 10, 2023}}

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:{ { unreferenced section|date=September 2020 } }

Γ d{{math|τ}}

File:All Pass Simulation.png

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Waveguides with certain symmetries may be solved using the method of separation of variables. Rectangular wave guides may be solved in rectangular coordinates.{{rp|143}} Round waveguides may be solved in cylindracal coordinates.{{rp|198}}

[https://en.wikipedia.org/w/index.php?title=Talk:Speed_of_electricity&curid=8817754&diff=839637350&oldid=839523951]

Talk:Speed_of_electricity#Incorrect section: Speed of electromagnetic waves in good conductors

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Talk:Displacement current

Talk:Displacement current#Untrue assumptions in the “Current in capacitors” sub section

{{Confusing section| reason=the known variables and the unknown variables of the system of equations is not clear. The rules given do not cover the case where two loops share nodes, but neither contains the other.}}

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curl E = -{{math|∂}}B/{{math|∂}}t

curl H = {{math|∂}}D/{{math|∂}}t + J_conduction + J_source = curl H = {{math|∂}}D/{{math|∂}}t + σE + J_source

:where σ = conductivity

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:$\backslash mathbf\{E\}\; =\; -\backslash mathbf\{\backslash nabla\}\backslash phi\; -\; \backslash frac\{\backslash partial\backslash mathbf\{A\}\}\{\backslash partial\; t\},\; \backslash ,$

Add alternate explanations of skin effect.

Add magnetic vector potential to the transformer page.

The solutions of the telegrapher's equations{{Citation |last=Hayt |first= William H. |year= 1989 |title= Engineering Electromagnetics |edition= 5th |publisher= McGraw-Hill |isbn= 0070274061|url= https://archive.org/details/engineeringelect5thhayt }}{{rp|381-392}} are functions of two parameters which are the propagation constant, $\backslash gamma$, and the characteristic impedance, $Z\_c$. The propagation constant is often written as $\backslash gamma\; =\; \backslash alpha\; +\; j\; \backslash beta$ where $\backslash alpha$ is called the attenuation constant and $\backslash beta$ is called the phase constant. The natural units for $\backslash alpha$ and $\backslash beta$ are nepers per unit length and radians per unit length, respectively.

There are two solutions which can be interpreted as a forward propagating wave and a reverse propagating wave.

The forward wave, which has a voltage of $V\_F(x)$ and a current of $I\_F(x)$, is given by:

:$V\_F(x)\; =\; V\_\{(F,0)\}\; e^\{-\backslash gamma\; x\}$

:$I\_F(x)\; =\; \backslash frac\; \{V\_F(x)\}\; \{Z\_c\}$

The reverse wave, which has a voltage of $V\_R(x)$ and a current of $I\_R(x)$, is given by:

:$V\_R(x)\; =\; V\_\{(R,d)\}\; e^\{-\backslash gamma\; (d-x)\}$

:$I\_R(x)\; =\; -\backslash frac\; \{V\_R(x)\}\{Z\_c\}$

where

:$\backslash gamma\; =\; \backslash alpha\; +\; j\; \backslash beta\; =\; \backslash sqrt\{(R\; +\; j\backslash omega\; L)(G\; +\; j\backslash omega\; C)\}$.{{rp|385}}

:$Z\_c\; =\; \backslash sqrt\{\backslash frac\; \{(R\; +\; j\; \backslash omega\; L)\}\; \{(G\; +\; j\backslash omega\; C)\}\}$.{{rp|385}}

:$v\; =\; \backslash frac\; \backslash omega\; \backslash beta$ = the propagation velocity.

:$\backslash lambda\; =\; \backslash frac\; \{2\; \backslash pi\}\; \backslash beta$ = wavelength.

:$d$ = length of the transmission line.

:$V\_\{(F,0)\}$ = a constant that depends on conditions at $x=0$.

:$V\_\{(F,d)\}$ = a constant that depends on conditions at $x=d$.

The negative sign in the expression for $I\_R(x)$ indicates that the current in the reverse wave is traveling in a direction that is opposite to the reference direction, which is the direction of the forward propagating wave. $Z\_c,\; \backslash gamma,\; \backslash alpha,\; \backslash beta,\; \backslash lambda,$ and $v$ are secondary parameters which means that they can be expressed in terms of the primary parameters $R,\; G,\; L,$ and $C$.

File:Newfoundland-Azores 1928 Submarine Telegraph Cable Estimated Velocity vs Frequency.png

File:Electron buildup movie from "Controlled double-slit electron diffraction" Roger Bach et al 2013 New J. Phys. 15 033018.gif

The propagation constant and the characteristic impedance may be factored as

:$Z\_c\; =Z\_0\; \backslash sqrt\{\backslash frac\; \{(1\; +\; r\_\backslash omega)\}\; \{(1+\; g\_\backslash omega)\}\}$ .

:$\backslash gamma\; =\; j\; \backslash frac\; \backslash omega\; \{v\_0\}\; \backslash sqrt\{(1\; +r\_\backslash omega)(1+\; g\_\backslash omega\; )\}$.

where

:$Z\_0\; =\backslash sqrt\{\; \backslash frac\; L\; C\; \}$ = nominal high frequency impedance.

:$v\_0\; =\backslash frac\; 1\; \backslash sqrt\{\; \{LC\}\; \}$ = nominal high frequency phase velocity.

:$r\_\backslash omega\; =\; \backslash frac\; \{R\}\; \{j\backslash omega\; L\}$ = impedance ratio.

:$g\_\backslash omega\; =\; \backslash frac\; \{G\}\; \{j\backslash omega\; C\}$ = admittance ratio.

where

The terms $v\_0$ and $Z\_0$ are relatively independent of frequency. The frequency dependent behavior of $\backslash gamma$ and $Z\_c$ can be separated into nine regions according to whether the ratios $g\_\backslash omega$ and $r\_\backslash omega$ are much less than unity, much greater than unity or about equal to unity. In practice, four of the combinations do not occur leaving five regions from DC to high frequency.

File:Typical Good Transmission Line Parameter Ratioes.png

File:Typical Good Transmission Line Velocity.png

The graph to the left shows the frequency dependent variation of $g\_\backslash omega$ and $r\_\backslash omega$ of a transmission line with a good dielectric such as high density polyethylene. Separation of the spectrum into regions has been arbitrarily set at the frequencies at which one of the ratios is either 0.1 or 10.

The five regions may be easily seen in the adjacent graph of velocity versus frequency of a transmission line with a good dielectric such as polyethylene:

From lowest to highest frequency the regions are:

• Near DC (NDC) where $\backslash frac\; \{G\}\; \{j\backslash omega\; C\}\; >>\; 1$ and $\backslash frac\; \{R\}\; \{j\backslash omega\; L\}\; >>\; 1$
• Very low frequency (VLF) where $\backslash frac\; \{G\}\; \{j\backslash omega\; C\}\; \backslash approx\; 1$ and $\backslash frac\; \{R\}\; \{j\backslash omega\; L\}\; >>\; 1$
• Low frequency (LF) where and $\backslash frac\; \{R\}\; \{j\backslash omega\; L\}\; >>\; 1$
• Intermediate frequency (IF) where and $\backslash frac\; \{R\}\; \{j\backslash omega\; L\}\; \backslash approx\; 1$
• High frequency (HF) where and

The high frequency regime is most associated with transmission line behavior. It includes Ethernet, most broadcasting, ordinary video, computer buses, the higher portions of the digital telephony spectrum ( HDSL, ADSL, VDSL).

:$Z\_c\; =\backslash sqrt\{\; \backslash frac\; L\; C\; \}\; \backslash sqrt\{\backslash frac\; \{(1\; +\; \backslash frac\; \{R\}\; \{j\backslash omega\; L\})\}\; \{(1+\; \backslash frac\; \{G\}\; \{j\backslash omega\; C\}\; )\}\}\; \backslash approx\; \backslash sqrt\{\; \backslash frac\; L\; C\; \}\; (1\; +\; \backslash frac\; \{R\}\; \{2j\backslash omega\; L\}\; -\; \backslash frac\; \{G\}\; \{2j\backslash omega\; C\}\; )$

:$\backslash gamma\; =\; j\backslash omega\; \backslash sqrt\{\; L\; C\; \}\; \backslash sqrt\{(1\; +\; \backslash frac\; \{R\}\; \{j\backslash omega\; L\})(1+\; \backslash frac\; \{G\}\; \{j\backslash omega\; C\}\; )\}\; \backslash approx\; j\backslash omega\; \backslash sqrt\{\; L\; C\; \}\; (1\; +\; \backslash frac\; \{R\}\; \{2j\backslash omega\; L\}\; +\; \backslash frac\; \{G\}\; \{2j\backslash omega\; C\}\; )\; =\; j\backslash omega\; \backslash sqrt\{\; L\; C\; \}\; +\; \backslash frac\; \{R\}\; \{2\}\backslash sqrt\{\; \backslash frac\; C\; L\; \}\; +\; \backslash frac\; \{G\}\; \{2\}\backslash sqrt\{\; \backslash frac\; L\; C\; \}$

:$\backslash alpha\; =\; \backslash frac\; \{R\}\; \{2\}\backslash sqrt\{\; \backslash frac\; C\; L\; \}\; +\; \backslash frac\; \{G\}\; \{2\}\backslash sqrt\{\; \backslash frac\; L\; C\; \}$

:$\backslash beta\; =\; \backslash omega\; \backslash sqrt\{\; L\; C\; \}$

:$v\; =\; \backslash frac\; \backslash omega\; \backslash beta\; =\; \backslash frac\; 1\; \backslash sqrt\{\; L\; C\; \}$

The intermediate frequency regime includes the lower portions of the digital telephony spectrum (ISDN, HDSL, ADSL, the upper frequencies of music,

The low frequency regime includes voice telephony, the lower frequencies of music,

The very low frequency regime includes

If $G\; \backslash equiv\; 0$ then there is no near DC regime. This is the case if the dielectric is vacuum.

The near DC regime expressions approach their DC values as frequency approaches zero, except wavelength which approaches infinity.

:$Z\_c\; =\; \backslash sqrt\{\backslash frac\; R\; G\}\; \backslash sqrt\{\; \backslash frac\; \{(1\; +\; \backslash frac\; \{j\backslash omega\; L\}\; R)\}\; \{(1\; +\; \backslash frac\; \{j\backslash omega\; C\}\; G)\}\; \}\; \backslash approx\; \backslash sqrt\{\backslash frac\; R\; G\}$.

:$\backslash gamma\; =\; \backslash sqrt\{R\; G\}\backslash sqrt\{(1\; +\; \backslash frac\; \{j\backslash omega\; L\}\; R)(1\; +\; \backslash frac\; \{j\backslash omega\; C\}\; G)\}\; \backslash approx\; \backslash sqrt\{R\; G\}(1\; +\; \backslash frac\; \{j\backslash omega\; L\}\; \{2R\}\; +\; \backslash frac\; \{j\backslash omega\; C\}\; \{2G\})$.

:$\backslash alpha\; =\; \backslash sqrt\{R\; G\}$.

:$\backslash beta\; =\; \backslash sqrt\{R\; G\}(\; \backslash frac\; \{\backslash omega\; L\}\; \{2R\}\; +\; \backslash frac\; \{\backslash omega\; C\}\; \{2G\})$. Typically $\backslash frac\; \{\backslash omega\; C\}\; \{2G\}\; >>\; \backslash frac\; \{\backslash omega\; L\}\; \{2R\}$ so $\backslash beta\; \backslash approx\; \backslash frac\; \{\backslash omega\; C\}\; \{2\}\; \backslash sqrt\{\backslash frac\; R\; G\}$.

:$v\; =\; \backslash frac\; \{2\}\; \{\; C\}\; \backslash sqrt\{\backslash frac\; G\; R\}$.

The near DC regime expressions approach their DC values as frequency approaches zero, except wavelength which approaches infinity.

:$Z\_\{c,dc\}\; =\; \backslash sqrt\{\backslash frac\; \{R\_\{dc\}\}\; \{G\_\{dc\}\}\}$.

:$\backslash alpha\_\{dc\}\; =\; \backslash sqrt\{\; R\_\{dc\}\; G\_\{dc\}\; \}$.

:$\backslash beta\_\{dc\}\; =\; 0$

:$v\_\{dc\}\; =\; \backslash frac\; \{2\}\; \{C\_\{dc\}\}\; \backslash sqrt\{\backslash frac\; \{G\_\{dc\}\}\; \{R\_\{dc\}\}\; \}$.

The following gallery shows the frequency dependence of some of the other secondary parameters.

File:Typical Good Transmission Line Wavelength.png|Wavelength of a typical RG-59 coaxial transmission line with a good (high resistivity) dielectric insulator in meters.

File:Typical Good Transmission Line Loss.png|Loss of a typical RG-59 coaxial transmission line with a good (high resistivity) dielectric insulator in dB/meter.

File:Typical Good Transmission Line Characteristic Impedance.png|Magnitude of the characteristic impedance of a typical RG-59 coaxial transmission line with a good (high resistivity) dielectric insulator in ohms.

File:Typical Good Transmission Line Characteristic Impedance Phase.png|Phase of the characteristic impedance of a typical RG-59 coaxial transmission line with a good (high resistivity) dielectric insulator in degrees.

File:Simple Time Domain Reflectometer Diagram.png|Simple TDR made from lab equipment

File:Time Domain Reflectometer made from common lab equipment.JPG|Simple TDR made from lab equipment

File:TDR trace of cable with open termination.JPG|TDR trace of a transmission line with an open termination

File:TDR trace of cable with a short circuit termination.JPG|TDR trace of a transmission line with a short circuit termination

File:TDR trace of cable with a capacitor termination.jpg|TDR trace of a transmission line with a 1nF capacitor termination

File:TDR trace of cable terminationed with its characteristic impedance.jpg|TDR trace of a transmission line with an almost ideal termination

File:TDR trace of cable with cable terminated on an oscilloscope.JPG|TDR trace of a transmission line terminated on an oscilloscope high impedance input. The blue trace is the pulse as seen at the far end. It is offset so that the baseline of each channel is visible

File:TDR Step Same Baseline.JPG|TDR trace of a transmission line terminated on an oscilloscope high impedance input driven by a step input from a matched source. The blue trace is the signal as seen at the far end.

File:Typical Good Transmission Line Velocity.png|Phase velocity of a typical RG-59 coaxial transmission line with a good (high resistivity) dielectric insulator relative to the speed of light in vacuum.

File:Typical Good Transmission Line Wavelength.png|Wavelength of a typical RG-59 coaxial transmission line with a good (high resistivity) dielectric insulator in meters.

File:Typical Good Transmission Line Loss.png|Loss of a typical RG-59 coaxial transmission line with a good (high resistivity) dielectric insulator in dB/meter.

File:Typical Good Transmission Line Characteristic Impedance.png|Magnitude of the characteristic impedance of a typical RG-59 coaxial transmission line with a good (high resistivity) dielectric insulator in ohms.

File:Typical Good Transmission Line Characteristic Impedance Phase.png|Phase of the characteristic impedance of a typical RG-59 coaxial transmission line with a good (high resistivity) dielectric insulator in degrees.

This involves a slight bit of synthesis from Brenner and Javid. Page 599 gives i₂/i₁ = 1/n and 1/n = n₁/n₂ which can be combined by simple arithmetic to give i₁n₁ = i₂n₂ or i₁n₁ - i₂n₂ = 0.

Hayt & Kemmerly in Engineering Circuit Analysis, 5'th on page 446 state plainly that i₁n₁ = i₂n₂, but it doesn't seem worth adding a reference for that.

(i₁n₁ - i₂n₂) is the magnetomotive force, using the current reference directions given in the figure in the same section.

First, an analogy. I can compute the velocity of my car by computing the time derivative of the number displayed on the odometer. That doesn’t mean the odometer causes the car’s velocity. The relationship holds because they are both caused by the same thing, which is the motor. Faraday's law of induction (FLI) states that the EMF (path integral of the electric field) in closed loop is proportional to the time derivative of the total flux the enclosed in the loop, including the flux in the core. This is a very useful relationship for designing transformers and predicting their behavior. This doesn’t mean that the flux in the core causes the EMF in the secondary, although it is often taught that way. Most of the time, that is good enough. But, in fact, if the flux in the core actually caused the EMF in the secondary, that would be action at a distance. Feynman makes this point in Volume 2 of the Feynman lectures, chap 15 section 5 in the second paragraph following equation 15.36.

Modern physicists have worked very hard to eliminate action at a distance. The modern formulation is that the currents produce the magnetic vector potential, A, at the wires. A produces a component of the electric field, E in accordance with E = -∂A/∂t (actually E= -{{math|∇}}φ -∂A/∂t, but I am ignoring φ). The line integral of -∂A/∂t over a closed path is the EMF. Faraday’s law of induction (FLI) works because B = {{math|∇ × }} A; E at the wires has the same cause as B in core. FLI is useful for engineers because they almost always have the transformer connected to a circuit which provides a complete path. However, E = -∂A/∂t, gives the E field at each infinitesimal part of the path.

I am not going to try to put this in the article; it is probably too technical. I’m going for WP:RIGHTGREATWRONGS, but I will try to edit the article so it is not in conflict with the modern formulation. For example, instead of “EMF is caused by the changing flux in the core” I may write “EMF is equal the rate of change of the flux in the core.” The first version is simpler and more direct, but it is a fiction whereas the second version is a correct statement.

Your comments are invited.

Apparently they knew this in 1913

https://books.google.com/books?id=71w2AQAAMAAJ&pg=PA947&lpg=PA947&dq=vector+potential+transformers&source=bl&ots=Sf4P9Qk63c&sig=n3sg0l_dPQ96fwIJT3tm-tzjF7g&hl=en&sa=X&ved=2ahUKEwjh9PGrgpLfAhWvna0KHTMNBrM4FBDoATAHegQIAxAB#v=onepage&q=vector%20potential%20transformers&f=false

And 1971

https://books.google.com/books?id=60EbAQAAMAAJ&pg=PA1195&dq=%22vector+potential%22+transformers&hl=en&sa=X&ved=0ahUKEwjk7KeAhJLfAhVKQ6wKHUKyDKAQ6AEIXzAJ#v=onepage&q=%22vector%20potential%22%20transformers&f=false

File:Magnetic flux around air-gap in transformer core made from E and I laminations.png

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File:Interleaved Transformer E-I Laminations.jpg

File:EI-transformer core interleaved with flux paths.png

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Files uploaded by Roy McCammon

File:Electric Transmission.png

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File:Gaussian Pulse in Transmission Line.png

File:Gaussian Pulse in a Two-Wire Transmission Line.png

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By definition, the diameter of #. 36 AWG is 0.005 inches, and # 0000 is 0.46 inches. The ratio of these diameters is 1:92, and there are 40 gauge sizes from #. 36 to # 0000, or 39 steps. The diameter of a # n AWG wire is determined, for gauges smaller than # 00 (36 to 0), according to the following formula:

:$d\_n\; =\; 0.005~\backslash mathrm\{inch\}\; \backslash times\; 92^\backslash frac\{36\; -\; n\}\{39\}\; =\; 0.127~\backslash mathrm\{mm\}\; \backslash times\; 92^\backslash frac\{36\; -\; n\}\{39\}$

The gauge can be calculated from the diameter using 

:$n\; =\; -39\backslash log\_\{92\}\; \backslash left(\; \backslash frac\{d\_n\}\{0.005~\backslash mathrm\{inch\}\}\; \backslash right)\; +\; 36$

:$n\; =\; \backslash frac\; \{-39\}\; \{\backslash log\_\{10\}\; 92\}\; \backslash log\_\{10\}\; \backslash left(\; \backslash frac\{d\_n\}\{0.005~\backslash mathrm\{inch\}\}\; \backslash right)\; +\; 36$

:$n\; =\; \backslash frac\; \{-39\}\; \{\backslash log\_\{e\}\; 92\}\; \backslash log\_\{e\}\; \backslash left(\; \backslash frac\{d\_n\}\{0.005~\backslash mathrm\{inch\}\}\; \backslash right)\; +\; 36$

In the ideal transformer section it notes that the ideal transformer is lossless, therefore power out = power in. If output volts go down then output current must go up. That’s a conclusion and not an explanation. Unfortunately, the reliable sources tend to write a bunch of equations and conclude that current transforms inversely with the turns ratio. Again it is a conclusion and not an explanation. I have an explanation, but without any reliable sources, it would be WP:OP. I’ll outline it here, hoping that maybe someone else can find a reliable source or determine that it is supportable by the sources already in the article. The gist is this, take a transformer with turns N_P and N_S and a load on the secondary and try to force a current I_P into the primary. If the quantity N_P × I_P – N_S × I_S ≠ 0, then you are forcing flux into the nominally infinite (or very large) self-inductance. It responds with an infinite (or very large) voltage seen on both the primary and the secondary. But there is a load on the secondary that would draw an infinite (or very large) current, so you can’t really get an infinite voltage. In fact, the secondary voltage that you can get is just enough to satisfy N_P × I_P – N_S × I_S = 0. The transformer, in effect, generates the voltage required to get the secondary current that will satisfy N_P × I_P – N_S × I_S = 0. A transformer with an infinite self-inductance is, so to speak, intolerant of N_P × I_P – N_S × I_S ≠ 0. You can sort of make a Lenz's law argument that any deviation from N_P × I_P – N_S × I_S = 0 would create huge reaction that would oppose the deviation.

:Sounds about right. I might have done it in terms of Ampere-Turns, as are commonly used in magnet design problems. Gah4 (talk) 06:19, 22 September 2016 (UTC)

The speed of electromagnetic waves in a low-loss dielectric is given by

: $\backslash quad\; v\; =\; \backslash frac\; \{1\}\; \backslash sqrt\{\backslash epsilon\; \backslash mu\}\; =\backslash frac\; \{c\}\; \backslash sqrt\{\backslash epsilon\_r\; \backslash mu\_r\}$ .{{Citation |last=Hayt |first= William H. |year= 1989 |title= Engineering Electromagnetics |edition= 5th |publisher= McGraw-Hill |isbn= 0070274061 }}{{rp|346}}

where

: $\backslash quad\; c$ = speed of light in vacuum.

: $\backslash quad\; \backslash mu\_0$ = the permeability of free space = 4π x 10⁻⁷ H/m.

: $\backslash quad\; \backslash mu\_r$ = relative magnetic permeability of the material. Usually in good dielectrics, $\backslash mu\_r\; =\; 1$.

: $\backslash quad\; \backslash mu$ = $\backslash mu\_r$ $\backslash mu\_0$.

: $\backslash quad\; \backslash epsilon\_0$ = the permeability of free space.

: $\backslash quad\; \backslash epsilon\_r$ = relative magnetic permeability of the material. Usually in good conductors, $\backslash epsilon\_r\; =\; 1$.

: $\backslash quad\; \backslash epsilon$ = $\backslash mu\_r$ $\backslash mu\_0$.

The speed of electromagnetic waves in a good conductor is given by

: $\backslash quad\; v\; =\; \backslash sqrt\{\; \backslash frac\; \{2\; \backslash omega\}\; \{\backslash sigma\; \backslash mu\}\}\; =\; \backslash sqrt\{\; \backslash frac\; \{4\; \backslash pi\}\; \{\backslash sigma\_c\; \backslash mu\_0\; \}\}\; \backslash sqrt\{\; \backslash frac\; \{\; f\; \}\; \{\backslash sigma\_r\; \backslash mu\_r\; \}\}\; \backslash approx\; \backslash left\; (\; 0.41\; \backslash frac\; \backslash mathsf\{m\}\; \backslash mathsf\{s\}\; \backslash right\; )\; \backslash sqrt\{\; \backslash frac\; \{\; f\; \}\; \{\backslash sigma\_r\; \backslash mu\_r\; \}\}$.{{rp|360}}

where

: $\backslash quad\; f$ = frequency.

: $\backslash quad\; \backslash omega$ = angular frequency = 2πf.

: $\backslash quad\; \backslash sigma\_c$ = conductivity of annealed copper = 5.96 x 10⁷ S/m.

: $\backslash quad\; \backslash sigma\_r$ = conductivity of the material relative to the conductivity of copper. For hard drawn copper $\backslash sigma\_r$ may be as low as 0.97.

: $\backslash quad\; \backslash sigma$ = $\backslash sigma\_r\; \backslash sigma\_c$.

In copper at 60 Hz, $v\; \backslash approx\; 3.2\; m/s$. Some sprinters can run twice as fast. As a consequence of Snell's Law and the extremely low speed, electromagnetic waves always enter good conductors in a direction that is normal to the surface, regardless of the angle of incidence. This velocity is then the speed with which electromagnetic waves penetrate into the conductor and is not the Drift velocity of the conduction electrons.

I have seven reliable sources of which two use plain language and the others simple formulas to state that the speed of an electromagnetic wave propagating in a good conductor is very much slower than the speed of light.

Plain language

:Hayt, (professor of electrical engineering emeritus at Purdue U.) {{Citation |last=Hayt |first= William H. |year= 1989 |title= Engineering Electromagnetics |edition= 5th |publisher= McGraw-Hill |isbn= 0070274061|url= https://archive.org/details/engineeringelect5thhayt }}{{rp|346}}, Engineering Electromagnetics, 5th, McGraw-Hill,1989 page 360 states "...velocity and wavelength within a good conductor ... For copper at 60 Hz, λ=5.36 cm and v= 3.22 m/s, ..."

:Kraus,(professor of electrical engineering and astronomy emeritus at Ohio State U.) Electromagnetics, page 451, Table 10-6 for copper gives velocity at 60Hz of 3.2 m/s

Simple formula{{Citation |last=Balanis |first= Constantine A. |year= 2012 |title= Engineering Electromagnetics |edition= 2nd |publisher= Wiley |isbn= 978-0-470-58948-9 }}{{rp|142}}

:Balanis,(professor of electrical engineering at Arizona State U.) Advanced Engineering Electromagnetics, 2 ed, 2012, Wiley, page 142, TABLE 4-1 gives the velocity in a good conductor as

: $v\; \backslash approx\; \backslash sqrt\{\; \{\; 2\; \backslash omega\}\; \backslash over\; \{\; \backslash mu\; \backslash sigma\}\; \}$

Rao{{Citation |last=Rao |first= Nannapaneni Narayanna |edition= 4th |year= 1994 |title= Elements of Engineering Electromagnetics |publisher= Prentice Hall |isbn= 0-13-948746-8 }}{{rp|332}},(professor of electrical and computer engineering at U. of Illinois at Urbana) Elements of Engineering Electromagnetics page 332 equation 6.81c

: $v\; \backslash approx\; \backslash sqrt\{\; \{\; 4\; \backslash pi\; f\}\; \backslash over\; \{\; \backslash mu\; \backslash sigma\}\; \}$

Kraus, Electromagnetics, page 450, equation 11

: $v\_c\; =\; \backslash sqrt\{\; \{\; 2\; \backslash omega\}\; \backslash over\; \{\; \backslash sigma\; \backslash mu\}\; \}$

Sadiku, Elements of Electromagnetics, 1989, p446, section 10.6 Plane waves in good conductors. equation (10.51)b

: $u\; =\; \backslash sqrt\{\; \{\; 2\; \backslash omega\}\; \backslash over\; \{\; \backslash mu\; \backslash sigma\}\; \}$

Taking

:: $\backslash omega\; =377$

:: $\backslash mu\; =4\; \backslash pi\; \backslash times\; 10^\{-7\}$

:: $\backslash sigma\; =5.96\; \backslash times\; 10^7$

yields

: $v\; \backslash approx\; 3.17\; m/s$

Two simple formulas{{Citation |last=Harrington |first= Roger F. |year= 1961 |title= Time-Harmonic Electromagnetic Fields |publisher= McGraw-Hill |isbn= 0-07-026745-6 }}{{rp|50-52}}

:Harrington,(professor of electrical engineering at Syracuse U.) Time-Harmonic Electromagnetic Fields, 1961, McGraw-Hill, page 52 gives

: $v\; =\; \{\; \{\; \backslash omega\}\; \backslash over\; \{\; k^\{\backslash prime\}\}\; \}$

and on page 50 gives for a good conductor

: $k^\{\backslash prime\}\; =\; \backslash sqrt\; \{\; \{\; \backslash omega\; \backslash mu\; \backslash sigma\; \}\; \backslash over\; \{2\}\; \}$

Using the previous values gives

: $k^\{\backslash prime\}\; =\; 118.81$

: $v\; =\; \{\; 377\; \backslash over\; 118.81\}\; =\; 3.17\; m/s$

Stratton,(professor of physics emeritus at MIT) Electromagnetic Theory, 1941, McGraw-Hill, page 522

gives the wavelength within a conductor as

: $\backslash lambda\; =\; 2\; \backslash pi\; \backslash sqrt\; \{\; \{\; 2\}\; \backslash over\; \{\backslash omega\; \backslash mu\; \backslash sigma\; \}\; \}$

and by common knowledge

: $v\; =\; f\; \backslash lambda\; =\; \{\; \backslash omega\; \backslash over\; \{2\; \backslash pi\}\}\; \backslash lambda$

yielding

: $v\; =\; \{\; \backslash omega\; \backslash over\; \{2\; \backslash pi\}\}\; 2\; \backslash pi\; \backslash sqrt\; \{\; \{\; 2\}\; \backslash over\; \{\backslash omega\; \backslash mu\; \backslash sigma\; \}\; \}\; =\; \backslash sqrt\; \{\; \{\; 2\; \backslash omega\}\; \backslash over\; \{\; \backslash mu\; \backslash sigma\; \}\; \}$

Note that Jackson cites Stratton, Harrington, Sadiku, and Kraus

Jackson 2nd page 337 and 3'rd page 354, eq. 8.9 and 8.10 give the phase term for a wave propagating into the conductor as $\{i\; \backslash xi\}\; \backslash over\; \backslash delta$ where ξ=depth into conductor and δ=skin depth.

Magnetic current is, nominally, a fictitious current composed of fictitious moving magnetic mono-poles. It has the dimensions of volts. Magnetic currents produce an electric field analogously to the production of a magnetic field by electric currents. Magnetic current density is usually represented by the symbol M, which has the units of v/m² (volts per square meter). A given distibution of electric change can be mathematically replaced by an equivalent distribution of magnetic current. This fact can be used to simplify some electromagnetic field problems.

{{efn| "For some electromagnetic problems, their solution can often be aided by the introduction of equivalent impressed electric and magnetic current densities."

{{Citation |last=Balanis |first= Constantine A. |year= 2012 |title= Advanced Engineering Electromagnetics |publisher= John Wiley |isbn= 978-0-470-58948-9 |doi= |p=3}}}}

{{efn| "there are many other problems where the use of fictitious magnetic currents and charges is very helpful." {{Citation |last=Jordan|first= Edward |last2=Balmain|first2=Keith G.|year= 1968 |title=Electromagnetic Waves and Radiating Systems|edition= 2nd |publisher= Prentice-Hall | lccn=68-16319 |isbn= |doi= | p=466}}}}

File:Magnetic Current Produces Electric Field By Left Hand Rule.svg

The direction of the electric field produced by magnetic currents is determined by the left-hand rule (opposite to the right-hand rule) as evidenced by the negative sign in the equation curl E = -M. One component of M is the familiar term {{math|∂}}B/{{math|∂}}t, which is referred to as the magnetic displacement current or more properly as the magnetic displacement current density.

{{efn| "Because of the symmetry of Maxwell's equations, the ɚB/ɚt term ... has been designated as a magnetic displacement current density." }}

{{efn| "interpreted as ... magnetic displacement current ..." }}

{{efn| "it also is convenient to consider the term ɚB/ɚt as a magnetic displacement current density." {{Citation |last=Harrington |first= Roger F. |year= 1961 |title= Time-Harmonic Electromagnetic Fields |edition= |publisher= McGraw-Hill |isbn=0-07-026745-6 |doi= |p=7}}}}

{{Citation |last=Balanis |first= Constantine A. |year= 1982 |title= Antenna Theory |publisher= John Wiley |isbn= 0471592684 |doi=}}

{{cite book| last=Temes|first= Gabor C. | last2= LaPatra| first2= Jack W.|year= 1977|title=Circuit Synthesis and Design |edition= |publisher= McGraw-Hill |isbn= 0-07-063489-0 }}{{rp|286}}

{{rp|286}}

{{efn| "Hence, the low sensitivity property of the doubly loaded LC ladder is preserved." }}{{rp|291}}

{{efn| "Hence, the low sensitivity property of the doubly loaded LC ladder is preserved." }}{{rp|291}}

$E\text{'}\; =\; E\; +\; \backslash frac\; \{v\}\; \{c\}\; \backslash times\; B$

$B\text{'}\; =\; B\; -\; \backslash frac\; \{v\}\; \{c\}\; \backslash times\; E$

$H\text{'}\; =\; H\; -\; v\; \backslash times\; D$

As late as 1963, Purcell offered the following low velocity transformations as suitable for calculating the electric field experienced by a jet plane tranvelling in the Earth's magnetic field.

:$E\text{'}\; =\; E\; +\; v\; \backslash times\; B$.

:$B\text{'}\; =\; B\; -\; \backslash epsilon\_0\; \backslash mu\_0\; v\; \backslash times\; E$. {{efn| Note: Purcell uses electrostatic units so the constants are different. This is the MKS version. {{Citation |last=Purcell |first= Edward M. |year= 1963 |title= Electricity and magnetism |edition= 1st |publisher= McGraw-Hill | LCCN=64-66016}}}}{{rp|222}}

In 1973 Bellac and Levy-Leblond state that these equations are incorrect or misleading because they do not correspond to any consistent Galilean limit. Rousseaux gives a simple example showing that a transformation from an initial inertial frame to a second frame with a speed of v₀ with respect to the first frame and then to a third frame moving with a speed v₁ with respect to the second frame would give a result different from going directly from the first frame to the third frame using a relative speed of (v₀ + v₁).

Bellac and Levy-Leblond offer two transformations that do have consistent Galilean limits as follows:

The electric limit applies when electric field effects are dominant such as when Faraday's law of induction was insignificant.

:$E\text{'}\; =\; E$.

:$B\text{'}\; =\; B\; -\; \backslash epsilon\_0\; \backslash mu\_0\; v\; \backslash times\; E$.

The magnetic limit applies when the magnetic field effects are dominant.

:$E\text{'}\; =\; E\; +\; v\; \backslash times\; B$.

:$B\text{'}\; =\; B$.

Statement by Germain Rousseaux: "For the experiments of electrodynamics of moving bodies with low speeds, the Galilean theory is the most adapted because it is easier of stake in work from the calculus point of view and does not bring in the kinematics effect of Special Relativity which are absolutely unimportant in the Galilean limit.{{cite journal

| title = Forty years of Galilean Electromagnetism (1973-2013)

| first = Germain

| last = Rousseaux

| journal = EPJ Plus

| issue = 128

| pages = 81

| publisher = The European Journal of Physics Plus (2013)

| location =

| date = August 2013

| url = http://www.pprime.fr/sites/default/files/pictures/pages-individuelles/D2/germain/EPJP2013.pdf

| issn =

| doi = 10.1140/epjp/i2013-13081-5

| id =

| accessdate = March 18, 2015}} {{rp|12}}.

:My 2 cents worth.

:*Volt is about the unit of measurement and should be a separate article.

:Everything else is tied together by the equation E = -∇(φ) -σA/σt, where φ is the electrodynamic retarded electric scalar potential and A is the retarded magnetic vector potential.

:*Voltage is, to me, a circuit quantity. It’s what a voltmeter would read between two points in a circuit. If the leads to the voltmeter went through a region where σA/σt ≠ 0, then the voltmeter reading would depend on the path of the leads to the voltmeter and the voltage would be ill defined.

:*Electromotive force, I usually associate with a path integral of the electric field (as defined above) tangential to a definite path. Usually the path follows a conductor. Typical paths may include a battery, generator, transformer, loop antenna, microphone, transducer. EMF for a specific path is unique. The path may be implicit. In a particular usage, there may be no mention of the path integral, but it is there never-the-less. If the beginning point and ending point of the path integral were in a continuous region where σA/σt = 0 then the voltage measured between those points by a voltmeter connected to those points and entirely in the region would read a unique value and EMF would numerically be the same as the voltage.

:* Potential has more than one meaning. The article will have to deal with that. For me, potential is a field quantity, although potential is used as a synonym for voltage in both DC and AC circuits.

::- The electrodynamic retarded scalar electric potential, which is used above in the equation for E. It is well defined everywhere except for points occupied by point charges. The difference in potential between two points is also well defined but may be physically meaningless. In general, the potential difference between two points is not the voltage between those two points.

::- The electrostatic potential that applies to electrostatics and DC circuits or any situation where σA/σt = 0. In this case, the difference in potential between two points is the same as the voltage between those two points and EMF evaluated along any path is also equal to the same voltage.{{Citation|last1=Bedell|first1=Frederick|title=History of A-C Wave Form, Its Determination and Standardization|journal=Transactions of the American Institute of Electrical Engineers|volume=61|issue=12|page=864|doi=10.1109/T-AIEE.1942.5058456}} "The magnetic flux is that flux which passes through any and every surface whose perimeter is the closed path" {{cite book| last = Hayt| first = William| title = Engineering Electromagnetics | edition = 5th| page = 312| isbn = 0-07-027406-1| publisher = McGraw-Hill| date = 1989}} {{citation|"Faraday's Law, which states that the electromotive force around a closed path is equal to the negative of the time rate of change of magnetic flux enclosed by the path"{{cite book|last=Jordan|first= Edward |last2=Balmain|first2=Keith G.| title = Electromagnetic Waves and Radiating Systems | edition = 2nd| page = 100| | publisher = Prentice-Hall| date = 1968}} "All the concepts of Chap. 5 translate verbatim to the transmission line case.", Sophocles J. Orfanidis, Electromagnetic Waves and Antennas; Chap. 8, "Transmission Lines" [http://www.ece.rutgers.edu/~orfanidi/ewa/ch11.pdf]; Chap. 5, "Reflection and Transmission" [http://www.ece.rutgers.edu/~orfanidi/ewa/ch05.pdf]

:If you use a definition of circuit like a roughly circular line, route, or movement that starts and finishes at the same place. That is, a mathematical definition, then path and circuit are similar. But in EE, circuit has a different definition. Gah4 (talk) 06:28, 22 September 2016 (UTC)

::Good morning Gah4. Thanks for your comments. I use my user page to prepare comments to be pasted into other pages, so they may not make any sense unless seen in the context of the target page. Also, I may edit them on the target page, but I don't come back here an echo the edits. So, what you find here may be incorrect, incomplete or obsolete and I don't make any attempt to maintain it. In this particular case, circuit means whatever it means in the target article, which was probably Faraday's law of induction. Comments and suggestions are welcome on that article's talk page. Cheers. Constant314 (talk) 14:27, 22 September 2016 (UTC)

:::That is what I suspected. Your talk page is on my watch list from a previous discussion, and so I happened to see it. The one on wire gauge is interesting. I never saw the formula listed before, and didn't even know that there was one. I wasn't looking in too much detail, though. Sometimes I am not ready for a real talk page discussion. Thanks, Gah4 (talk) 14:48, 22 September 2016 (UTC)

_________________________________________________________

{{see also|List of physical quantities|List of electromagnetism equations}}

Electromagnetic units are part of a system of electrical units based primarily upon the magnetic properties of electric currents, the fundamental SI unit being the ampere. The units are:

In the electromagnetic cgs system, electric current is a fundamental quantity defined via Ampère's law and takes the permeability as a dimensionless quantity (relative permeability) whose value in a vacuum is unity. As a consequence, the square of the speed of light appears explicitly in some of the equations interrelating quantities in this system.

________________________________________________________________________________________________________________

{{Hidden|Some math|

:$$

\frac{\partial^2}{{\partial x}^2} I =

L C \frac{\partial^2}{{\partial t}^2} I +

(R C + G L) \frac{\partial}{\partial t} I + G R I

}}

{{Hidden begin

|style = border: 1px dashed gray; width: 50%;background: #ffffff;

|titlestyle = padding-right:5em;

|showhide=left

|expanded=true

|titlestyle

|title=  useful math

}}

$x=y$

{{Hidden end}}

________________________________________________________________________________________________________________

Testing

Talk:Gyroscope#Another animation

These work.

Electric current#Conventions

near-field

:Causality#Engineering

:Causality#Physics

:Fermi energy#Related quantities

:Wikipedia:Causality#Fields#Science#Engineering

:Wikipedia:Make technical articles understandable

:Wikipedia:Make technical articles understandable#Rules of thumb

:Wikipedia:Make technical articles understandable#Write one level down

These do not work. They link to the top of the article.

:Causality#Fields#Science

:Wikipedia:Make technical articles understandable#Rules of thumb_Write one level down

:Wikipedia:Make technical articles understandable#(Rules of thumb#Write one level down)

:Wikipedia:Make technical articles understandable#{Rules of thumb#Write one level down}

:Wikipedia:Make technical articles understandable#Rules of thumb(Write one level down)

:Wikipedia:Make technical articles understandable#Rules of thumb[Write one level down]

:Wikipedia:Make technical articles understandable[Rules of thumb#Write one level down]

:Wikipedia:Make technical articles understandable#Rules of thumb##Write one level down

:Wikipedia:Make technical articles understandable#Rules of thumb.Write one level down

:Wikipedia:Make technical articles understandable#Rules of thumb/Write one level down

:Wikipedia:Make technical articles understandable#Rules of thumb\Write one level down

________________________________________________________________________________________________________________

{{Citation | last=Hayt |first=William |last2= Kemmerly |first2= Jack E. | title = Engineering Circuit Analysis |edition= 5th | year = 1993 | publisher= McGraw-Hill |isbn=007027410X }}

{{Citation |last= Reitz |first= John R. |last2= Milford |first2= Frederick J. |last3= Christy |first3= Robert W. |year= 1993 |title= Foundations of Electromagnetic Theory|edition= 4th |publisher= Addison-Wesley |isbn= 0201526247}}

________________________________________________________________________________________________________________

Talk:Telegrapher's equations#Solutions of the Telegrapher's Equations as Circuit Components

this talk page comment

________________________________________________________________________________________________________________

: $Z\_\backslash mathsf\; S\; =\; Z\_\backslash mathsf\; L^*\; \backslash ,$

: $Z\_\backslash mathrm\{load\}\; =\; Z\_\backslash mathrm\{source\}^*\; \backslash ,$

: $Z\_\backslash text\{load\}\; =\; Z\_\backslash text\{source\}^*\; \backslash ,$

Alt text is useful for the visually impaired. The following equation has alt text.

: $Z\_\backslash mathsf\{load\}\; =\; Z\_\backslash mathsf\{source\}^*\; \backslash ,$

One of the best usenet discussions on Poynting vector and wires and angular momentum. Some day I'm going to write it up as a dialog between the Tortoise, Achilles and some other characters.

• [http://groups.google.com/group/sci.physics/browse_frm/thread/ab9e9181deca125f/ccd9181e9cf6bb84?hl=en&tvc=1&q=sci.physics.*++rbmccammon#ccd9181e9cf6bb84 A Usenet discussion of Poynting vector and wires and angular momentum]

• [http://www.eetimes.com/design/microwave-rf-design/4200760/SPICE-Simulation-of-Transmission-Lines-by-the-Telegrapher-s-Method-Part-1-of-3-?Ecosystem=microwave-rf-design SPICE Simulation of Transmission Lines]

• [https://www.edn.com/tear-down-the-propagating-plane-wave/ Depicting the plane wave with other useful fields]

A discussion on usenet about the self discharge time constant for some tpyes of capacitors, including polypropylene.

https://groups.google.com/forum/#!searchin/sci.electronics.design/bruhns$20capacitor%7Csort:relevance/sci.electronics.design/SRzTxKyPXVE/JoOBSYFUjn4J

See{{cite web | title = Sign Conventions in Electromagnetic (EM) Waves | url = https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-007-electromagnetic-energy-from-motors-to-lasers-spring-2011/readings/MIT6_007S11_sign.pdf}}The function $f(x)\; =\; \backslash cos(2\; \backslash pi\; \backslash xi\_0\; x)\; \backslash equiv\; \backslash cos(-2\; \backslash pi\; \backslash xi\_0\; x)$ is also a signal with frequency $\backslash xi\_0$, but the integral obviously produces identical responses at both $\backslash xi\_0$ and $-\backslash xi\_0$, which is also consistent with Euler's formula: $\backslash cos(2\; \backslash pi\; \backslash xi\_0\; x)\; \backslash equiv\; \backslash tfrac\{1\}\{2\}\; e^\{i\; 2\; \backslash pi\; \backslash xi\_0\; x\}\; +\; \backslash tfrac\{1\}\{2\}\; e^\{-i\; 2\; \backslash pi\; \backslash xi\_0\; x\}.$.

Consider the functions:

::$u(t)\; =\; 0.5\; \backslash cos(\; 2\; \backslash pi\; f\; t\; )\; +\; 0.866\; \backslash sin(\; 2\; \backslash pi\; f\; t\; )\; \backslash ,$

::$v(t)\; =\; 0.866\; \backslash cos(\; 2\; \backslash pi\; f\; t\; )\; +\; 0.5\; \backslash sin(\; 2\; \backslash pi\; f\; t\; )\; \backslash ,$

::In mathematics, $cos(\; 2\; \backslash pi\; f\; t\; )\backslash ,$ and $sin(\; 2\; \backslash pi\; f\; t\; )\backslash ,$ are considered to be basis vectors of a two dimensional vector space. The function $v(t)\; \backslash ,$ would be represented by a point with coordinates (0.866, 0.5). The line connecting the origin to this point makes a positive angle of 30 degrees with respect to the origin. Therefore in mathematics, the phase of $v(t)\; \backslash ,$ would be 30 degrees. Likewise, $u(t)\; \backslash ,$ would be represented by (.5,0.866) and its phase would be 60 degrees. The phase difference between $u(t)\; \backslash ,$ and $v(t)\; \backslash ,$ would be 30 degrees.

In engineering, a sinusoid has a negative phase shift with respect to some other sinusoid, if the peaks of the first sinusoid occurs after the peaks of the second sinusoid.

Using a trig identity, $u(t)\; \backslash ,$ and $v(t)\; \backslash ,$ can be written as:

::$u(t)\; =\; cos(\; 2\; \backslash pi\; f\; t\; -\; \backslash pi/3\; )\; \backslash ,$ note: $\backslash pi\; /3\; =\; \backslash ,$ 60 degrees.

::$v(t)\; =\; cos(\; 2\; \backslash pi\; f\; t\; -\; \backslash pi/6\; )\; \backslash ,$ note: $\backslash pi\; /6\; =\; \backslash ,$ 30 degrees.

::The peaks of $v(t)\; \backslash ,$ and $u(t)\; \backslash ,$ occur after the peak of $cos(\; 2\; \backslash pi\; f\; t\; )\; \backslash ,$ by 30 and 60 degrees respectively. Therefore in engineering, the phase of $v(t)\; \backslash ,$ and $u(t)\; \backslash ,$ would be -30 degrees and -60 degrees respectively. The phase difference of $u(t)\; \backslash ,$ with respect to $v(t)\; \backslash ,$ would be -30 degrees meaning that the peaks of $u(t)\; \backslash ,$ occur after the peaks of $v(t)\; \backslash ,$ by 30 degrees .

In most cases, consistantly using the changing the sign of the phase only changes the sign of the imaginary part of the computation. In othere words, the result of using one convention or the other produces results that are conjugates of each other.

One place this shows up is in the definition of the Fourier transform.

$x(t)=\; \backslash frac\; \{1\}\; \{2\; \backslash pi\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; X(\backslash omega)e^\{j\; \backslash omega\; t\}\; d\backslash omega$

• from Oppenheim, Alan V.; Willsky, Alan S.; Young, Ian T. (1983), Signals and Systems (1st ed.), Prentice-Hall, {{ISBN|0138097313}}

The following use the same convention:

• Gregg, W. David (1977), Analog & Digital Communication, John Wiley, {{ISBN|0471326615}}
• Stein, Seymour; Jones, J. Jones (1967), Modern Communnication Principles, McGraw-Hill, page 4, equation 1-5
• Hayt, William; Kemmerly, Jack E. (1971), Engineering Circuit Analysis (2nd ed.), McGraw-Hill, {{ISBN|0070273820}}, page 535, equation 8b.

$h(t)=\; \backslash frac\; \{1\}\; \{2\; \backslash pi\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; e^\{-j\; \backslash omega\; t\}\; H(\backslash omega)d\backslash omega$

• from Press, William H.; Teukolsky, Saul A.; Vetterling, William T. (2007), Numerical Recipes (3rd ed.), Cambridge University Press, {{ISBN|9780521880688}}, page 692.

The following use the same convention:

• Jackson, John Davd (1999), Classical Electrodynamics (3rd ed.), John-Wiley, {{ISBN|047130932X}}, page 372, equation 8.89
• Stratton, Julius Adams (1941), Electromagnetic Theory, McGraw-Hill page 294, equation 47
• Reitz, John R.; Milford, Frederick J.; Christy, Robert W. (1993), Foundations of Electromagnetic Theory, Addison-Wesley, {{ISBN|0201526247}}, page 607, equation VI-2

I've looked in 11 references and found two ways to write down the equation for a plane wave.

(In all cases I have changed i to j for consistancy.)

The following group use a form for the plane wave that involves $-\; \backslash omega\; t\; \backslash ,$ such as $e^\{j(\backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{x\}\; -\; \backslash omega\; t)\; \}\; \backslash ,$ or $\backslash cos(kx-\backslash omega\; t)$.

• Griffiths, David (1989), Introduction to Electrodynamics, Prentice-Hall, {{ISBN|013481374X}}, page 356, equation 8.60
• Jackson, John Davd (1999), Classical Electrodynamics (3rd ed.), John-Wiley, {{ISBN|047130932X}}, page 296, equation 7.8
• Reitz, John R.; Milford, Frederick J.; Christy, Robert W. (1993), Foundations of Electromagnetic Theory, Addison-Wesley, {{ISBN|0201526247}}, page 416, equation 17-7

The following group use a form for the plane wave that involves $+\; \backslash omega\; t\; \backslash ,$ such as $e^\{j(\backslash omega\; t\; -\; \backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{x\})\; \}\; \backslash ,$ or $\backslash cos(\backslash omega\; t-kx)$.

• Crawford, Frank S. (1968), Waves - berkeley phisics course - volume 3, McGraw-Hill, page, page 333, equation 2-1
• Harrington, Roger F. (1961), Time-Harmonic Electromagnetic Fields, McGraw-Hill, page 39, equation 2-13
• Hayt, William (1989), Engineering Electromagnetics (5th ed.), McGraw-Hill, {{ISBN|0070274061}}, page 338, equation 6
• Jordan, Edward; Balmain, Keith G. (1968), Electromagnetic Waves and Radiating Systems (2nd ed.), Prentice-Hall, page 124,
• Marshall, Stanley V. (1987), Electromagnetic Concepts & Applications (1st ed.), Prentice-Hall, {{ISBN|0132490048}}, page 320, equation 12
• Ramo, Simon; Whinnery, John R.; van Duzer, The odore (1965), Fields and Waves in Communication Electronics, John Wiley, page 247, equation 10
• Sadiku, Matthew N. O. (1989), Elements of Electromagnetics (1st ed.), Saunders College Publishing, ISBN 993013846, page 432 equation 10.4
• Kraus, John D. (1984), Electromagnetics (3rd ed.), McGraw-Hill, {{ISBN|0070354235}}, page 385, equation 29.

I think this list is enough to establish that there is a group that uses $\backslash omega\; t\; \backslash ,$ and a group that uses $-\; \backslash omega\; t\; \backslash ,$

The first group are physicist. The second group, except for Crawford, are engineers. Kraus is an engineer but says he can use either convention.

So what is the difference? Not much, because at the end of the computation you discard the imaginary part and keep the real part. You wind up with terms of either $\backslash cos(\backslash omega\; t\; -\; \backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{x\})\backslash ,$ or $\backslash cos(\backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{x\}\; -\; \backslash omega\; t)\backslash ,$ which are equal.

________________________________________________________________________________________________________________

:The terms leading and lagging are used by engineers to describe the phase difference of one sinusoid to another. If the peaks of one sinusoid occur after the peaks of a second sinusoid, then the first sinusoid is said to lag the second sinusoid. If the peaks of the first sinusoid occur before peaks of the second sinusoid, then the first sinusoid is said to lead the second sinusoid.

::For example, sin(t) lags cos(t) by 90 degrees and cos(t) leads sin(t) by 90 degrees. Although sin(t) could be said to lead cos(t) by 270 degrees it is conventional to choose leading or lagging so that the angle is less than or equal to 180 degrees.

:In electric power distribution, leading and lagging power factor always refers to current relative to voltage. Thus a leading power factor means the current is leading the voltage.

________________________________________________________________________________________________________________

{{Elvis Presley singles}}

Feynman{{Harvtxt|Feynman|1964|p=15_15}} and Jackson {{Harvtxt|Jackson|1999|p=246}} give the following integral equations for calculating the electric scalar potential, $\backslash phi\; \backslash ,$ and the magnetic vector potential, $\backslash mathbf\{A\}\backslash ,$ at point $p\_1\backslash ,$ and time $t\backslash ,$ from the current density distribution and charge density distribution. $\backslash mathbf\{A\}\backslash ,$ is a 3 dimensional vector. The notation differs slightly from both sources.

$\backslash mathbf\{A\}(p\_1,t)=\; \{\backslash mu\_0\; \backslash over\; 4\; \backslash pi\}\; \backslash int\_\{V\_2\}\{\; \backslash mathbf\{j\}\; (p\_2,t\text{'})\backslash over\; r\_\{12\}\}\; dV\; \backslash ,$

$\backslash phi\; (p\_1,t)=\; \{1\; \backslash over\; 4\; \backslash pi\; \backslash epsilon\_0\; \}\; \backslash int\_\{V\_2\}\{\; \backslash rho\; (p\_2,t\text{'})\backslash over\; r\_\{12\}\}\; dV\; \backslash ,$

:: where

::: $p\_1\backslash ,$ is the point at which the value of $\backslash mathbf\{A\}\backslash ,$ and $\backslash phi\; \backslash ,$ are to be calculated.

::: $t\backslash ,$ is the time at which the value of $\backslash mathbf\{A\}\backslash ,$ and $\backslash phi\backslash ,$ are to be calculated.

::: $p\_2\backslash ,$ is a point at which the value of $\backslash mathbf\{j\}\backslash ,$ or $\backslash rho\; \backslash ,$ or both are non-zero.

::: $t\text{'}\; =\; t\; -\; \{\; r\_\{12\}\; \backslash over\; c\; \}\; \backslash ,$ is a time earlier than $t\backslash ,$ by $\{\; r\_\{12\}\; \backslash over\; c\; \}\backslash ,$ which is the time it takes an effect generated at $p\_2\backslash ,$ to propagate to $p\_1\backslash ,$ at the speed of light. $t\text{'}\backslash ,$ is also called retarded time.

::: $\backslash mathbf\{A\}(p\_1,t)\backslash ,$ is the magnetic vector potential at point $p\_1\backslash ,$ and time $t\; \backslash ,$.

::: $\backslash phi(p\_1,t)\backslash ,$ is the electric scalar potential at point $p\_1\backslash ,$ and time $t\backslash ,$.

::: $\backslash mathbf\{j\}(p\_2,t\text{'})\; \backslash ,$ is the current density at point $p\_2\backslash ,$ and time $t\text{'}\backslash ,$

::: $\backslash rho(p\_2,t\text{'})\; \backslash ,$ is the charge density at point $p\_2\backslash ,$ and time $t\text{'}\backslash ,$

:::$r\_\{12\}\backslash ,$ is the distance from point $p\_1\backslash ,$ to point $p\_2\backslash ,$

:::$V\_2\backslash ,$ is the volume of all points $p\_2\backslash ,$ where $\backslash mathbf\{j\}\backslash ,$ or $\backslash rho\; \backslash ,$ is non-zero at least sometimes.

: $\backslash mathbf\{A\}\backslash ,$ and $\backslash phi\; \backslash ,$ calculated in this way will satisfy with the condition: $\backslash nabla\; \backslash cdot\; \backslash mathbf\{A\}\; +\; \{\; c^2\; \backslash partial\; \backslash phi\; \backslash over\; \backslash partial\; t\; \}\; =\; 0\; \backslash ,$

There are a few notable things about equation for $\backslash mathbf\{A\}\backslash ,$. First, the position of the source point $p\_2\backslash ,$ only enters the equation as a scalar distance from $p\_1\backslash ,$ to $p\_2\backslash ,$. The direction from $p\_1\backslash ,$ to $p\_2\backslash ,$ does not enter into the equation. The only thing that matters about a source point is how far away it is. Second, the integrand uses retarded time. This simply reflects the fact that changes in the sources propagate at the speed of light. And third, the equation is a vector equation. In Cartesian coordinates, the equation separates into three equations thus{{Harvtxt|Kraus|1984|p=189}}:

$\backslash mathbf\{A\_x\}(p\_1,t)=\; \{\backslash mu\_0\; \backslash over\; 4\; \backslash pi\}\; \backslash int\_\{V\_2\}\{\; \backslash mathbf\{j\_x\}\; (p\_2,t\text{'})\backslash over\; r\_\{12\}\}\; dV\; \backslash ,$ where $\backslash mathbf\{A\_x\}\backslash ,$ and $\backslash mathbf\{j\_x\}\backslash ,$ are the components of $\backslash mathbf\{A\}\backslash ,$ and $\backslash mathbf\{j\}\backslash ,$ in the direction of the x axis.

$\backslash mathbf\{A\_y\}(p\_1,t)=\; \{\backslash mu\_0\; \backslash over\; 4\; \backslash pi\}\; \backslash int\_\{V\_2\}\{\; \backslash mathbf\{j\_y\}\; (p\_2,t\text{'})\backslash over\; r\_\{12\}\}\; dV\; \backslash ,$

$\backslash mathbf\{A\_z\}(p\_1,t)=\; \{\backslash mu\_0\; \backslash over\; 4\; \backslash pi\}\; \backslash int\_\{V\_2\}\{\; \backslash mathbf\{j\_z\}\; (p\_2,t\text{'})\backslash over\; r\_\{12\}\}\; dV\; \backslash ,$

In this form it is easy to see that the component of $\backslash mathbf\{A\}\backslash ,$ in a given direction depends only on the components of $\backslash mathbf\{j\}\backslash ,$ that are in the same direction. If the current is carried in a long straight wire, the $\backslash mathbf\{A\}\backslash ,$ points in the same direction as the wire.

not posted

7. $\backslash mathbf\{B\}=\; \backslash nabla\; \backslash times\; \backslash mathbf\{A\}\; \backslash ,$ {{Harvtxt|Jackson|1999|p=239}}

8. $\backslash mathbf\{E\}=\; -\backslash nabla\; \backslash phi\; -\; \{\; \backslash partial\; \backslash mathbf\{A\}\; \backslash over\; \backslash partial\; t\; \}\; \backslash ,$

When magnetic effects are dominent, equation 8 can be simplified to:

9. $\backslash mathbf\{E\}=\; -\; \{\; \backslash partial\; \backslash mathbf\{A\}\; \backslash over\; \backslash partial\; t\; \}\; \backslash ,$

Consider two long straight zero (or very low) resistance wires extending along the x axis. One carries a sinusoidally varying current that produces a magnetic vector potential that is directed in the same direction (along the x axis). The electric field in the second wire has the opposite direction to the magnetic vector potential. So the current in one long wire tends to produce a current of in the opposite direction in a parallel wire.

Terman {{Harvtxt|Terman|1943|p=112}} "The power factor ... tends to be independent of frequency, since the fraction of energy lost during each cycle ... is substantially independent of the number of cycles per second, over wide frequency ranges."

Terman is using older terminology. Power factor in a capacitor is the same as dissipation factor. The comment also applies to the loss tangent of a dielectric.

Griffiths {{Harvtxt|Griffiths|1989|p=223}}, regarding the calculation of the magnitude of the B field in a toroidal inductor "determining its magnitude is ridiculously easy."

Halliday {{Harvtxt|Halliday|1962|p=901}} regarding the calculation of the magnitude of the B field in a toroidal inductor "For a close-packed coil and no iron nearby ..." .

Time-Harmonic Electromagnetic Fields, McGraw-Hill, 1961, Reprinted 1987

Page 63, "From the field theory point of view, this is equivalent to assuming that no EZ or HZ exists. Such a wave is called transverse electromagnetic, abbreviated TEM. This is not the only wave possible on a transmission line, for Maxwell's equations show that infinitely many wave types can exist. Each possible wave is called a mode, and a TEM wave is called a transmission line mode. All other waves, which must have an EZ or an HZ or both, are called higher-order modes. The higher-order modes are usually important only in the vicinity of a feed point, or in the vicinity of a discontinuity on the line."

Page 147, "We therefore conjecture that all wave functions can be expressed as superposition of plane waves".

Electromagnetic Theory, McGraw-Hill, 1941

Page 533, "The transport of energy along the cylinder takes place entirely in the external dielectric. The internal energy surges back and forth and supplies the Joule heat losses."

Dreams of a Final Theory, Pantheon Books, 1992

Chapter 6. Beautiful Theories:

:Page 142, "Just as the electromagnetic force between two electrons is due in quantum mechanics to the exchange of photons, the force between photons and electrons is due to the exchange of electrons."

Chapter 7: Against Philosophy

:Page 170, " ... Einstein's special theory of relativity in effect banished the ether and replaced it with empty space as the medium that carries electromagnetic impulses."

Engineering Electromagnetics, McGraw-Hill, 1989

Chapter 11: The Uniform Plane Wave,

Section 5: Propagation in Good Conductors: the Skin Effect

Page 360, "Electromagnetic energy is not transmitted in the interior of a conductor; it travels in the region surrounding the conductor, while the conductor merely guides the waves. The currents established at the conductor surface propagate into the conductor in a direction perpendicular to the current density, and they are attenuated by ohmic losses. This power loss is the price exacted by the conductor for acting as a guide."

Feynman{{Harvtxt|Feynman|1985|p=9}} regarding QED, "...you're not going to be able to understand it. ... my physics students don't understand it either. That's because I don't understand it. Nobody does."

Feynman{{Harvtxt|Feynman|1964|p=27_6}} regarding The ambiguity of field energy, "... but we must say that we do not know for certain what is the actual location in space of the electromagnetic field energy."

Feynman{{Harvtxt|Feynman|1964|p=27_8}} regarding Examples of energy flow, "As another example, we ask what happens in a piece of resistance wire when it is carrying a current. ... . . There is a flow of energy into the wire all around. It is, of course, equal to the energy being lost in the wire in the form of heat. So our crazy theory says that the electrons are getting their energy to generate heat because of the energy flowing into the wire from the field outside. Intuition would seem to tell us that the electrons get their energy from being pushed along the wire, so the energy should be flowing down (or up) along the wire. But the theory says that the electrons are really being pushed by an electric field, which has come from some charges very far away, and that the electrons get their energy for generating heat from these fields. The energy somehow flows from the distant charges into a wide area of space and then inward to the wire.

... that the energy is flowing into the wire from the outside, rather than along the wire. "

Feynman{{Harvtxt|Feynman|1964|p=15_7}} regarding real fields, "What we really mean by a real field is this: a real field is a mathematical function we use for avoiding the idea of action at a distance."

Feynman{{Harvtxt|Feynman|1964|p=15_8}} regarding real fields, "We have introduced A because ... it is ... a real physical field in the sense that we described above."

Feynman{{Harvtxt|Feynman|1964|p=15_8}} regarding The vector potential and quantum mechanics, "In our sense then, the A-field is real. ... The B-field in the whisker acts at a distance."

Standard Handbook for Electrical Engineers, 11th Edition, Fink, Donald G. editor, McGraw-Hill

Chapter 2, Section 40,

Page 2-13, "The energies stored in the fields travel with them, and this phenomenon is the basic and sole mechanism whereby electric power transmission takes place. Thus the electrical energy transmitted by means of transmission lines flows through the space surrounding the conductors, the latter (conductors) acting merely as guides.

"The usually accepted view that the conductor current produces the magnetic field surrounding it must be displaced by the more appropriate one that the electromagnetic field surrounding the conductor produces, through a small drain on its energy supply, the current in the conductor. Although the value of the latter (current) may be used in computing the transmitted energy, one should clearly recognize that physically this current produces only a loss and in no way has a direct part in the phenomenon of power transmission."

Ether and the Theory of Relativity, address on May 05, 1920 at University of Leyden

p6

"More careful reflection teaches us, however, that the special theory of relativity does not compel us to deny ether. We may assume the existence of an ether; only we must give up ascribing a definite state of motion to it, ...

According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light"

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Examples of annotated links

• {{annotated link|Voltage}}
• {{annotated link|Leapfrog filter}}