Elektromagnit maydon va zaryadlangan zarralar sistemasining umumiy impuls oʻzgarishi qonuni quyidagicha ifodalanadi:
![{\displaystyle {\dfrac {d}{dt}}\left({\textbf {G}}_{m}+{\textbf {G}}_{s}\right)=\oint {\textbf {T}}_{n}d\sigma ;\ \ \ \ \ (1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5cf32177b0db64bde4a358c7680ca5d8d22237)
bu yerda:
![{\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}};\ \ \ \ \ (2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c6138dd7bc9005493cab9a6c6f6b47c958d71a4)
boʻlib, normal orti n boʻlgan birlik yuzachaga taʼsir qiluvchi elektromagnit maydon kuchini ifodalaydi va sirtga taʼsir qiluvchi elektromagnit kuchlar zichligi deyiladi. Normalning yoʻnalishiga qarab Tn vektor ham turlicha boʻlishi mumkin. Normalning orti n sifatida, masalan, Ox oʻqining orti i olinsa, koʻramizki, Tx vektor shu oʻqqa perpendikulyar qoʻyilgan birlik yuzachaga taʼsir qiluvchi maydon kuchini ifodalaydi. Shuning kabi Ty va Tz vektorlar Oy va Oz oʻqlariga mos ravishda perpendikulyar boʻlgan birlik yuzachalarga taʼsir qiluvchi maydon kuchlarini ifodalaydi.
Sirtga taʼsir qiluvchi elektromagnit kuchlar zichligi Tn vektorning aniqlanishi uchun (2) ga muvofiq Tx, Ty,Tz vektorlar maʼlum boʻlishi lozim: bu vektorlarning tashkil etuvchilarini quyidagicha yozamiz:
![{\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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6327087a26f369c870b85892414539c4eba3c4a3)
![{\displaystyle T_{xy}={\frac {1}{4\pi }}\left(E_{x}E_{y}+H_{x}H_{y}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19bdd72af6817e4fa410030bbda48bc8534c083e)
![{\displaystyle T_{xz}={\frac {1}{4\pi }}\left(E_{x}E_{z}+H_{x}H_{z}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c739ec43bf4ff5aeb978ae42e8687ed2ba7881f)
![{\displaystyle T_{yx}={\frac {1}{4\pi }}\left(E_{y}E_{x}+H_{y}H_{x}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/414d61e9b1058687690ba2c22ad702e79de9269f)
![{\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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e02fcbe7a5d382b22dd3cb30fd984c9bce4a713)
![{\displaystyle T_{yz}={\frac {1}{4\pi }}\left(E_{y}E_{z}+H_{y}H_{z}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1528648efb861b9c420128a58db0045e07adb57a)
![{\displaystyle T_{zx}={\frac {1}{4\pi }}\left(E_{z}E_{x}+H_{z}H_{x}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8522fadf4a16257f50999de303e291bba41d791)
![{\displaystyle T_{zy}={\frac {1}{4\pi }}\left(E_{z}E_{y}+H_{z}H_{y}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d0621efa40b45edf7a3aea58eb7505345004866)
;
Agar
desak, koordinatalarning umumiy koʻrinishi
boʻladi, bu yerda
indeks 1, 2, 3,
qiymatlarni qabul qiladi. U vaqtda (3) ifodani quyidagi koʻrinishda yozib koʻrsatish mumkin:
![{\displaystyle T_{\alpha \beta }={\frac {1}{4\pi }}\left(E_{\alpha }E_{\beta }+H_{\alpha }H_{\beta }\right)-\delta _{\alpha \beta }{\frac {1}{8\pi }}\left(E^{2}+H^{2}\right);\ \ \ \ \ (4)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8765d59ae8c441131d480af075512b645f449826)
bu yerda
indekslar 1, 2, 3 qiymatlarga ega boʻlib,
esa Kroneker belgisi deyiladi va quyidagicha taʼriflanadi:
![{\displaystyle \delta _{\alpha \beta }={\begin{cases}0,\ {\textrm {agar}}\ \alpha \neq \beta \\1,\ {\textrm {agar}}\ \alpha =\beta \end{cases}};\ \ \ \ \ (5)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b088e5acb9b431cb935853aa5ac033c9c68da240)
(4) — tenglamadan maʼlumki,
![{\displaystyle T_{\alpha \beta }=T_{\beta \alpha };\ \ \ \ \ (6)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/414d4df3f39ac6091cac4b081c7bd9a1c93e2a5f)
Normal orti n bilan ixtiyoriy i
orasidagi burchak kosinusini
orqali belgilasak:
![{\displaystyle C_{n\beta }=\cos({\textbf {n}}^{\wedge }{\textbf {i}}_{\beta })=\left({\textbf {n}}\ {\textbf {i}}_{\beta }\right)\ \ \ \ \ (7)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc2ccf67207978eb9f6aef959dff388efa0c2198)
u holda
![{\displaystyle {\textbf {n}}=\sum C_{n\beta }{\textbf {i}}_{\beta };\ \ \ \ \ (8)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9afb8cae0d961c11229e2c44913150497767dc34)
Endi bizni qiziqtirayotgan Tn vektor uchun berilgan (2) ifodani kiritilgan belgilashlar asosida yozamiz:
![{\displaystyle {\textbf {T}}_{n}=\sum \left({\textbf {T}}_{\alpha }\ {\textbf {n}}\right){\textbf {i}}_{\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34c1d8867fc1147da7b43dc4438b07249ad958e1)
yoki (8) ga muvofiq
![{\displaystyle {\textbf {T}}_{n}=\sum \sum C_{n\beta }\left({\textbf {T}}_{\alpha }{\textbf {i}}_{\beta }\right){\textbf {i}}_{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c09e6922346da0f4b6082b8ca7514534e28eab8a)
Skalyar koʻpaytma
esa
vektorning
ort yoʻnalishidagi
tashkil etuvchisidir. Demak,
![{\displaystyle {\textbf {T}}_{n}=\sum \sum C_{n\beta }{\textbf {T}}_{\alpha \beta }{\textbf {i}}_{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b50979b6b2735a9b3f02cb7f7a9da2f259c822a)
yoki (6) ga muvofiq,
![{\displaystyle {\textbf {T}}_{n}=\sum \sum C_{n\beta }{\textbf {T}}_{\beta \alpha }{\textbf {i}}_{\alpha };\ \ \ \ \ (9)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f031636fd99bd256b221b781d40bccd05bb9ea28)
Berilgan
ortlar sistemasini koordinatalar boshi atrofida aylantirish natijasida kelib chiqqan yangi sistema ortlarini
orqali belgilaylik, bu yerda
. Normal orti n ixtiyoriy boʻlganligidan, n oʻrniga i'lolinishi mumkin. U vaqtda (9) ga muvofiq,
![{\displaystyle {\textbf {T}}'_{l}=\sum \sum C_{l\beta }{\textbf {T}}_{\beta \alpha }{\textbf {i}}_{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c8fac5fcf808d4d1b5866531cc513c58df02427)
Bu tenglikning ikki tomonini i'm (
) ortga skalyar ravishda koʻpaytirilsa,
![{\displaystyle T'_{lm}\sum \sum C_{l\beta }C_{m\alpha }{\textbf {T}}_{\beta \alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c5c4f54091f97308de34776823c2d4056fe897)
yokki
va
oʻrinlarini almashtirilsa,
![{\displaystyle T'_{lm}\sum \sum C_{l\alpha }C_{m\beta }{\textbf {T}}_{\alpha \beta };\ \ \ \ \ \ (10)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e171281beb732700c1020e30271e14ddabae673)
Mana shu almashtirish qonuniga boʻysungan
miqdorlar toʻplami ikkinchi rangli tenzor deyiladi. (4) da ifodalangan toʻqqizta
miqdorlar toʻplami elektromagnit impuls oqimi zichligining tenzori deb yuritiladi.
- R.X.Mallin, Klassik elektrodinamika, Oʻqituvchi, T., 1974