ⓘ Maxwellning elektromagnit kuchlanishlar tenzori. Vaqt birligidagi impuls ozgarishi tasir qiluvchi kuchga tengdir. Demak, ∂ ∂ t g m + g s = div 2 T ; 1 {\display ..

                                     

ⓘ Maxwellning elektromagnit kuchlanishlar tenzori

Vaqt birligidagi impuls ozgarishi tasir qiluvchi kuchga tengdir. Demak,

∂ ∂ t g m + g s = div 2 T ; 1 {\displaystyle {\frac {\partial }{\partial t}}\left{\textbf {g}}_{m}+{\textbf {g}}_{s}\right={\textrm {div}}^{2}{\textbf {T}};\ \ \ \ \ \ \ 1}

tenglamaga muvofiq, hajmda tasir qiluvchi elektromagnit kuchlar zichligi div 2 T {\displaystyle {\textrm {div}}^{2}{\textbf {T}}} bilan boglangan boladi. U vaqtda,

∮ T n d σ = ∫ div 2 T d V ; 2 {\displaystyle \oint {\textbf {T}}_{n}d\sigma =\int {\textrm {div}}^{2}{\textbf {T}}dV;\ \ \ \ \ \ \ 2}

ifodadan korish mumkinki, hajmda tasir qiluvchi elektromagnit kuchlarni shu hajmni chegaralovchi yopiq sirtga tasir qiluvchi elektromagnit kuchlar bilan almashtirish mumkin. Shu sababli elektromagnit impuls oqimining zichlik tenzori 2 T {\displaystyle ^{2}{\textbf {T}}} Maxwellning elektromagnit kuchlanishlar tenzori deb ham yuritiladi. Bu tenglamaning komponentlari:

T x = 1 4 π E x 2 + H x 2 − 1 8 π E 2 + H 2, {\displaystyle T_{xx}={\frac {1}{4\pi }}\leftE_{x}^{2}+H_{x}^{2}\right-{\frac {1}{8\pi }}\leftE^{2}+H^{2}\right,} T x y = 1 4 π E x E y + H x H y, {\displaystyle T_{xy}={\frac {1}{4\pi }}\leftE_{x}E_{y}+H_{x}H_{y}\right,} T x z = 1 4 π E x E z + H x H z, {\displaystyle T_{xz}={\frac {1}{4\pi }}\leftE_{x}E_{z}+H_{x}H_{z}\right,} T y x = 1 4 π E y E x + H y H x, {\displaystyle T_{yx}={\frac {1}{4\pi }}\leftE_{y}E_{x}+H_{y}H_{x}\right,} T y = 1 4 π E y 2 + H y 2 − 1 8 π E 2 + H 2, {\displaystyle T_{yy}={\frac {1}{4\pi }}\leftE_{y}^{2}+H_{y}^{2}\right-{\frac {1}{8\pi }}\leftE^{2}+H^{2}\right,} T y z = 1 4 π E y E z + H y H z, {\displaystyle T_{yz}={\frac {1}{4\pi }}\leftE_{y}E_{z}+H_{y}H_{z}\right,} T z x = 1 4 π E z E x + H z H x, {\displaystyle T_{zx}={\frac {1}{4\pi }}\leftE_{z}E_{x}+H_{z}H_{x}\right,} T z y = 1 4 π E z E y + H z H y, {\displaystyle T_{zy}={\frac {1}{4\pi }}\leftE_{z}E_{y}+H_{z}H_{y}\right,} T z = 1 4 π E z 2 + H z 2 − 1 8 π E 2 + H 2 ; 3 {\displaystyle T_{zz}={\frac {1}{4\pi }}\leftE_{z}^{2}+H_{z}^{2}\right-{\frac {1}{8\pi }}\leftE^{2}+H^{2}\right;\ \ \ \ \ 3} ;

Maxwell tenzori uchun

T n = T x n i + T y n j + T z n k ; 4 {\displaystyle {\textbf {T}}_{n}={\textbf {T}}_{x}{\textbf {n}}{\textbf {i}}+{\textbf {T}}_{y}{\textbf {n}}{\textbf {j}}+{\textbf {T}}_{z}{\textbf {n}}{\textbf {k}};\ \ \ \ \ 4}

ga asosan quyidagilarni yozish mumkin:

T n x = T x n x + T x y n y + T x z n z, {\displaystyle T_{nx}=T_{xx}n_{x}+T_{xy}n_{y}+T_{xz}n_{z},} T n y = T y x n x + T y n y + T y z n z, {\displaystyle T_{ny}=T_{yx}n_{x}+T_{yy}n_{y}+T_{yz}n_{z},} T n z = T z x n x + T z y n y + T z n z, 5 {\displaystyle T_{nz}=T_{zx}n_{x}+T_{zy}n_{y}+T_{zz}n_{z},\ \ \ \ \ \ \ 5}
                                     

1. Maxwellning elektr kuchlanishlar tenzori

Elektr maydon yonalishi x oqining yonalishi bilan mos tushsin 1-rasm. Kuchlanganlik vektori E {\displaystyle {\textbf {E}}} va elementar yuzacha normalining n {\displaystyle {\textbf {n}}} orti qogoz sirtida yotsin, ular orasidagi burchak φ {\displaystyle \varphi } boladi, y oqi, demak, j {\displaystyle {\textbf {j}}} ort qogoz sirtiga perpendikulyar qilib olinsa, z oqi, yani k {\displaystyle {\textbf {k}}} ort rasmda korsatilganidek yonalgan boladi.

Agar faqatgina elektr maydongina mavjud deb qaralsa, u holda:

E x = E, E y = 0, E z = 0, {\displaystyle E_{x}=E,\ \ \ E_{y}=0,\ \ \ E_{z}=0,} H x = 0, H y = 0, H z = 0 ; 6 {\displaystyle H_{x}=0,\ \ \ H_{y}=0,\ \ \ H_{z}=0;\ \ \ \ \ \ \ 6}

Normal ortining komponentlari esa

n x = cos ⁡ φ, n y = 0, n z = sin ⁡ φ ; 7 {\displaystyle n_{x}=\cos \varphi,\ \ \ n_{y}=0,\ \ \ n_{z}=\sin \varphi ;\ \ \ \ \ \ \ 7}

Maxwell tenzori komponentlari 6 va 7 ga muvofiq:

T x = E 2 8 π, T x y = 0, T x z = 0, {\displaystyle T_{xx}={\frac {E^{2}}{8\pi }},\ \ \ T_{xy}=0,\ \ \ T_{xz}=0,} T y x = 0, T y = E 2 8 π, T x z = 0, {\displaystyle T_{yx}=0,\ \ \ T_{yy}={\frac {E^{2}}{8\pi }},\ \ \ T_{xz}=0,} T z x = 0, T z y = 0, T z = E 2 8 π ; 8 {\displaystyle T_{zx}=0,\ \ \ T_{zy}=0,\ \ \ T_{zz}={\frac {E^{2}}{8\pi }};\ \ \ \ \ \ \ 8}

Demak, elektr maydon yonalishiga perpendikulyar qoyilgan elementar yuzacha birligiga maydon boylab tasir qiluvchi kuch, yani normal kuchlanish T x {\displaystyle T_{xx}} bilan ifodalanadi va maydon energiyasi zichligi bilan olchanadi. Elektr maydon yonalishid yotgan elementar yuza birligiga uning normaliga qarama qarshi yonalishda tasir qiluvchi kuch, yani bosim T y {\displaystyle T_{yy}} yoki T z {\displaystyle T_{zz}} bilan ifodalanadi va maydon energiyasi zichligi bilan olchanadi.

6 va 7 ga muvofiq 3 ni quyidagicha yozish mumkin:

T n x = E 2 8 π cos ⁡ φ, T n y = 0, T n z = − E 2 8 π sin ⁡ φ ; 9 {\displaystyle T_{nx}={\frac {E^{2}}{8\pi }}\cos \varphi,\ \ \ T_{ny}=0,\ \ \ T_{nz}=-{\frac {E{2}}{8\pi }}\sin \varphi ;\ \ \ \ \ \ 9}

demak,

| T n = E 2 8 π ; 10 {\displaystyle |{\textbf {T}}_{n}={\frac {E^{2}}{8\pi }};\ \ \ \ \ \ \ 10}

yani elementar yuza birligiga tasir qiluvchi maydon kuchining son qiymati maydon energiyasining zichligiga teng, ammo maydon yonalishining elementar yuzachaga nisbatan qandayligiga bogliq emas.

Elektr kuchlanishlar tenzori bilan magnit kuchlanishlar tenzori yigindisi elektromagnit kuchlanishlar tenzorini hosil qiladi