Maxwellning elektromagnit kuchlanishlar tenzori
Vaqt birligidagi impuls oʻzgarishi taʼsir 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 taʼsir qiluvchi elektromagnit kuchlar zichligi div ( 2 ) T {\displaystyle {\textrm {div}}^{(2)}{\textbf {T}}} bilan bogʻlangan boʻladi. 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 koʻrish mumkinki, hajmda taʼsir qiluvchi elektromagnit kuchlarni shu hajmni chegaralovchi yopiq sirtga taʼsir 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 x = 1 4 π ( E x 2 + H x 2 ) − 1 8 π ( E 2 + H 2 ) , {\displaystyle T_{xx}={\frac {1}{4\pi }}\left(E_{x}^{2}+H_{x}^{2}\right)-{\frac {1}{8\pi }}\left(E^{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 }}\left(E_{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 }}\left(E_{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 }}\left(E_{y}E_{x}+H_{y}H_{x}\right),} T y y = 1 4 π ( E y 2 + H y 2 ) − 1 8 π ( E 2 + H 2 ) , {\displaystyle T_{yy}={\frac {1}{4\pi }}\left(E_{y}^{2}+H_{y}^{2}\right)-{\frac {1}{8\pi }}\left(E^{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 }}\left(E_{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 }}\left(E_{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 }}\left(E_{z}E_{y}+H_{z}H_{y}\right),} T z 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 }}\left(E_{z}^{2}+H_{z}^{2}\right)-{\frac {1}{8\pi }}\left(E^{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 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 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 z n z ,               ( 5 ) {\displaystyle T_{nz}=T_{zx}n_{x}+T_{zy}n_{y}+T_{zz}n_{z},\ \ \ \ \ \ \ (5)}