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Area Radio wave propagation / Radiation, paths and velocity of electromagnetic waves

IEV ref 705-02-10

en
complex Poynting vector
for a sinusoidal wave of angular frequency ω, the vector product:

1 2 E _ × H _ MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meWabaGacmGadiWaaiWabaabaiaafaaakeaadaWcaaqaaiaaigdaaeaacaaIYaaaamaaFiaabaWaaWaaaeaacaWGfbaaaaGaay51GaGaey41aq7aa8HaaeaadaadaaqaaiaadIeaaaaacaGLxdcadaahaaWcbeqaaiabgEHiQaaaaaa@4367@

when the electric field vector E (t) MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meWabaGacmGadiWaaiWabaabaiaafaaakeaadaWhcaqaaiaadweaaiaawEniaiaacIcacaWG0bGaaiykaaaa@3E5E@ and the magnetic field vector H (t) MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meWabaGacmGadiWaaiWabaabaiaafaaakeaadaWhcaqaaiaadweaaiaawEniaiaacIcacaWG0bGaaiykaaaa@3E5E@ are represented at the same point, in complex notation, by the equations:

E (t)=Re( E _ e jωt ) MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meqabaGacmGadiWaaiqabaabaiaafaaakeaadaWhcaqaaiaadweaaiaawEniaiaacIcacaWG0bGaaiykaiabg2da9iGackfacaGGLbGaaGPaVlaacIcadaadaaqaamaaFiaabaGaamyraaGaay51GaaaaiaaykW7ciGGLbWaaWbaaSqabeaacaGGQbGaeqyYdCNaamiDaaaakiaacMcaaaa@4CF4@

H (t)=Re( H _ e jωt ) MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meqabaGacmGadiWaaiqabaabaiaafaaakeaadaWhcaqaaiaadweaaiaawEniaiaacIcacaWG0bGaaiykaiabg2da9iGackfacaGGLbGaaGPaVlaacIcadaadaaqaamaaFiaabaGaamyraaGaay51GaaaaiaaykW7ciGGLbWaaWbaaSqabeaacaGGQbGaeqyYdCNaamiDaaaakiaacMcaaaa@4CF4@

where E _ MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meWabaGacmGadiWaaiWabaabaiaafaaakeaadaWhcaqaamaamaaabaGaamyraaaaaiaawEniaaaa@3C1C@ and H _ MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meWabaGacmGadiWaaiWabaabaiaafaaakeaadaWhcaqaamaamaaabaGaamyraaaaaiaawEniaaaa@3C1C@ are not time-dependent and are generally complex, H _ MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meWabaGacmGadiWaaiWabaabaiaafaaakeaadaWhcaqaamaamaaabaGaamisaaaaaiaawEniamaaCaaaleqabaGaey4fIOcaaaaa@3D3B@ being the complex conjugate of H _ MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meWabaGacmGadiWaaiWabaabaiaafaaakeaadaWhcaqaamaamaaabaGaamyraaaaaiaawEniaaaa@3C1C@

Note 1 – The real part of the complex Poynting vector is the average of the Poynting vector during one cycle.

Note 2 – The imaginary part of the complex Poynting vector is a vector of which, with certain reservations, the direction may be considered as being that of reactive energy propagation, and the magnitude may be regarded as the reactive power flux per unit surface area perpendicular to this direction, with a sign assigned conventionally.


fr
vecteur de Poynting complexe, m
en régime sinusoïdal de pulsation ω, produit vectoriel:

1 2 E _ × H _ MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meWabaGacmGadiWaaiWabaabaiaafaaakeaadaWcaaqaaiaaigdaaeaacaaIYaaaamaaFiaabaWaaWaaaeaacaWGfbaaaaGaay51GaGaey41aq7aa8HaaeaadaadaaqaaiaadIeaaaaacaGLxdcadaahaaWcbeqaaiabgEHiQaaaaaa@4367@

lorsque le vecteur champ électrique E (t) MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meWabaGacmGadiWaaiWabaabaiaafaaakeaadaWhcaqaaiaadweaaiaawEniaiaacIcacaWG0bGaaiykaaaa@3E5E@ et le vecteur champ magnétique H (t) MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meWabaGacmGadiWaaiWabaabaiaafaaakeaadaWhcaqaaiaadweaaiaawEniaiaacIcacaWG0bGaaiykaaaa@3E5E@ sont représentés au même point, en notation complexe, par les expressions:

E (t)=Re( E _ e jωt ) MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meqabaGacmGadiWaaiqabaabaiaafaaakeaadaWhcaqaaiaadweaaiaawEniaiaacIcacaWG0bGaaiykaiabg2da9iGackfacaGGLbGaaGPaVlaacIcadaadaaqaamaaFiaabaGaamyraaGaay51GaaaaiaaykW7ciGGLbWaaWbaaSqabeaacaGGQbGaeqyYdCNaamiDaaaakiaacMcaaaa@4CF4@

H (t)=Re( H _ e jωt ) MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meqabaGacmGadiWaaiqabaabaiaafaaakeaadaWhcaqaaiaadweaaiaawEniaiaacIcacaWG0bGaaiykaiabg2da9iGackfacaGGLbGaaGPaVlaacIcadaadaaqaamaaFiaabaGaamyraaGaay51GaaaaiaaykW7ciGGLbWaaWbaaSqabeaacaGGQbGaeqyYdCNaamiDaaaakiaacMcaaaa@4CF4@

E _ MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meWabaGacmGadiWaaiWabaabaiaafaaakeaadaWhcaqaamaamaaabaGaamyraaaaaiaawEniaaaa@3C1C@ et H _ MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meWabaGacmGadiWaaiWabaabaiaafaaakeaadaWhcaqaamaamaaabaGaamyraaaaaiaawEniaaaa@3C1C@ sont indépendants du temps généralement complexes, H _ MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meWabaGacmGadiWaaiWabaabaiaafaaakeaadaWhcaqaamaamaaabaGaamisaaaaaiaawEniamaaCaaaleqabaGaey4fIOcaaaaa@3D3B@ désignant le vecteur complexe conjugué de H _ MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdi9WrpeeC0lXdi9qqqj=hEeeu0lXdbba9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meWabaGacmGadiWaaiWabaabaiaafaaakeaadaWhcaqaamaamaaabaGaamyraaaaaiaawEniaaaa@3C1C@

Note 1 – La partie réelle du vecteur de Poynting complexe est la moyenne du vecteur de Poynting pendant une période.

Note 2 – La partie imaginaire du vecteur de Poynting complexe est un vecteur dont on peut, sous certains réserves, considérer la direction comme celle de la propagation de l'énergie réactive, et le module, affecté d'un signe conventionnel, comme le flux de la puissance réactive par unité de surface perpendiculaire à cette direction.


ar
متجه بوينتنج المركب

de
komplexer Poynting-Vektor, m

es
vector de Poynting complejo

it
vettore complesso di Poynting

ko
복소 포인팅 벡터

ja
複素ポインティングベクトル

pl
wektor Poyntinga zespolony

pt
vector de Poynting complexo

sv
komplex poyntingvektor

Publication date: 1995-09
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