Elektromagnit toʻlqinlar quyidagi differensial tenglamalarga boʻysunadi:

{\displaystyle {\textrm {div}}{\textbf {A}}+{\frac {1}{c}}{\dfrac {\partial \varphi }{\partial t}}=0;\ \ \ \ \ \ (1)}
{\displaystyle \Delta {\textbf {A}}-{\frac {1}{c^{2}}}{\dfrac {\partial ^{2}{\textbf {A}}}{\partial t^{2}}}=-4\pi {\frac {1}{c}};\ \ \ \ \ \ (2)}
{\displaystyle \Delta \varphi -{\frac {1}{c^{2}}}{\dfrac {\partial ^{2}\varphi }{\partial t^{2}}}=-4\pi \rho ;\ \ \ \ \ \ \ (3)}
Koʻrinib turibdiki, xususiy hosilalardan iborat chiziqli differensial tenglamalar sistemasi bilan amal qilishga toʻgʻri keladi.

  

Cheksiz kichik hajmdagi zaryad {\displaystyle \delta e=\rho dV} boʻladi. Koordinatalar boshini shu zaryadda joylashgan deb hisoblaymiz. U vaqtda, {\displaystyle dV} hajmdan tashqaridagi {\displaystyle (R\neq 0)} nuqtalarda zaryad yoʻq, demak (2 — 3) ga muvofiq

{\displaystyle {\textrm {div}}{\textbf {A}}+{\frac {1}{c}}{\dfrac {\partial \varphi }{\partial t}}=0;\ \ \ \ \ \ (4)}
{\displaystyle \Delta {\textbf {A}}-{\frac {1}{c^{2}}}{\dfrac {\partial ^{2}{\textbf {A}}}{\partial t^{2}}}=0;\ \ \ \ \ \ (5)}
{\displaystyle \Delta \varphi -{\frac {1}{c}}{\dfrac {\partial ^{2}\varphi }{\partial t^{2}}}=0;\ \ \ \ \ \ (6)}
{\displaystyle \delta e} zaryadning {\displaystyle dv} hajmdan tashqaridagi nuqtalarda hosil qilgan maydoni koordinatalar boshigacha boʻlgan masofa va vaqtgagina bogʻliq, yaʼni maydon sferik simmetriyaga ega. Endi Laplas operatorini sferik koordinatalarda yozib koʻrsak, (6) ga muvofiq

{\displaystyle {\frac {1}{R^{2}}}{\dfrac {\partial }{\partial R}}\left(R^{2}{\frac {\partial \varphi }{\partial R}}\right)-{\frac {1}{c^{2}}}{\dfrac {\partial ^{2}\varphi }{\partial t^{2}}}=0;\ \ \ \ \ \ (7)}
Quyidagi koʻrinishda yangi funksiya kiritamiz:

{\displaystyle \varphi (t)={\frac {\psi (R,t)}{R}};\ \ \ \ \ \ (8)}
Koʻrinib turibdiki,

{\displaystyle {\frac {\partial \varphi }{\partial R}}={\frac {1}{R}}{\frac {\partial \psi }{\partial R}}-{\frac {\psi }{R^{2}}},}
{\displaystyle R^{2}{\frac {\partial \varphi }{\partial R}}=R{\frac {\partial \psi }{\partial R}}-\psi ,}
{\displaystyle {\dfrac {\partial }{\partial R}}\left(R^{2}{\frac {\partial \varphi }{\partial R}}\right)=R{\frac {\partial ^{2}\psi }{\partial R^{2}}}+{\frac {\partial \psi }{R}}-{\frac {\partial \psi }{R}}=R{\frac {\partial ^{2}\psi }{\partial R^{2}}},}
{\displaystyle {\frac {1}{R^{2}}}{\dfrac {\partial }{\partial R}}\left(R^{2}{\frac {\partial \varphi }{\partial R}}\right)={\frac {1}{R}}{\frac {\partial ^{2}\psi }{\partial R^{2}}}}
{\displaystyle {\frac {1}{c^{2}}}{\dfrac {\partial ^{2}\varphi }{\partial t^{2}}}={\frac {1}{c^{2}}}{\frac {\partial ^{2}\psi }{\partial t^{2}}}}
Demak, (7) ga muvofiq

{\displaystyle {\dfrac {\partial ^{2}\psi }{\partial R^{2}}}-{\frac {1}{c}}{\dfrac {\partial ^{2}\psi }{\partial t^{2}}}=0;\ \ \ \ \ \ (9)}
boʻladi. Bu yassi elektromagnit to'lqin tenglamasidir. Shunday qilib,

{\displaystyle \psi =\psi \_{1}\left(t+{\frac {R}{c}}\right)+\psi \_{2}\left(t-{\frac {R}{c}}\right).\ \ \ \ \ \ \ (10)}
To'lqin tenglamaning xususiy yechimigina bizni qiziqtiradi. Shu sababli {\displaystyle \psi \_{2}} bilan cheklanamiz:

{\displaystyle \psi =\psi \left(t-{\frac {R}{c}}\right).\ \ \ \ \ \ (11)}
U holda, (8) ga muvofiq

{\displaystyle \varphi (t)={\frac {\psi \left(t-{\frac {R}{c}}\right)}{R}}.\ \ \ \ \ \ (12)}
boʻladi. Koʻrinib turibdiki, vaqt oʻtishi bilan ({\displaystyle t>0}) toʻlqin radiusning orta borish tomoniga qarab tarqaladi. Nomaʼlum {\displaystyle \psi \left(t-{\frac {R}{c}}\right)} funksiyani aniqlash lozim.

Zaryadga cheksiz yaqin turgan nuqtalar uchun {\displaystyle {\frac {R}{c}}} cheksiz kichik. Demak, {\displaystyle \varphi (t)={\frac {\psi (t)}{R}}}. Maʼlumki, zaryadning Kulon potensiali:

{\displaystyle \varphi (t)={\frac {\delta e(t)}{R}}={\frac {\rho (t)dV}{R}}}
Demak, {\displaystyle \psi (t)=\rho (t)dV}, bu yerdan {\displaystyle \psi \left(t-{\frac {R}{c}}\right)=\rho \left(t-{\frac {R}{c}}\right)dV}, u vaqtda (12) ga muvofiq

{\displaystyle \varphi (t)={\frac {\rho \left(t-{\frac {R}{c}}\right)dV}{R}}.\ \ \ \ \ \ (13)}
Koʻramizki, (13) ga asosan, kuzatish nuqtasida vaqtning {\displaystyle t} momentidagi potensial vaqtning oldingi {\displaystyle t^{\prime }=t-{\frac {R}{c}}} momentidagi zaryad zichligi bilan aniqlanadi. Zaryad turgan joyda vaqtning {\displaystyle t^{\prime }=t-{\frac {R}{c}}} momentida paydo boʻlgan potensial {\displaystyle R} masofani {\displaystyle {\frac {\textbf {R}}{c}}} vaqtda oʻtib, kuzatish nuqtasiga {\displaystyle {\frac {R}{c}}} vaqt kechikish bilan yetib keladi. Shuning uchun (13) bilan ifodalangan potensial kechikuvchi potensial deb yuritiladi.

Zaryadlar sistemasining potensiali
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Zaryadlar sistemasining potensialini aniqlash uchun (13) formulani sistemaning barcha zaryadlari joylashgan hajm boʻyicha [integral]]lash lozim:

{\displaystyle \varphi (t)=\int {\frac {\rho \left(t-{\frac {R}{c}}\right)}{R}}\,dV;\ \ \ \ \ \ (14)}
Xuddi shuningdek,

{\displaystyle {\textbf {A}}(t)={\frac {1}{c}}\int {\frac {{\textbf {j}}\left(t-{\frac {R}{c}}\right)}{R}}\,dV;\ \ \ \ \ \ (15)}
Ushbu tenglamalarda zaryad turgan nuqtani koordinatalar boshi sifatida olingan. Agar koordinatalar boshini sistemaning ichidagi boshqa biror nuqtaga koʻchirilsa, yangi koordinatalar boshiga nisbatan zaryad turgan nuqtaning radius-vektori {\displaystyle {\textbf {r}}^{\prime }} va kuzatish nuqtasining radius-vektori {\displaystyle {\textbf {r}}} bilan belgilanadi. Demak,

{\displaystyle {\textbf {R}}={\textbf {r}}-{\textbf {r}}^{\prime };\ \ \ \ \ \ (16)}
boʻladi. Yangi koordinatalar boshiga nisbatan kechikuvchi potensiallar quyidagicha yoziladi:

{\displaystyle \varphi ({\textbf {r}},t)=\int {\frac {\rho \left({\textbf {r}}^{\prime },\ t-{\frac {|{\textbf {r}}-{\textbf {r}}^{\prime }|}{c}}\right)}{|{\textbf {r}}-{\textbf {r}}^{\prime }|}}\,dV;\ \ \ \ \ \ (17)}
{\displaystyle {\textbf {A}}({\textbf {r}},t)={\frac {1}{c}}\int {\frac {{\textbf {j}}^{\prime }\left({\textbf {r}}^{\prime },\ t-{\frac {|{\textbf {r}}-{\textbf {r}}^{\prime }|}{c}}\right)}{|{\textbf {r}}-{\textbf {r}}^{\prime }|}}\,dV;\ \ \ \ \ \ (18)}

Adabiyotlar
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* *R.X.Mallin*, Klassik elektrodinamika, Toshkent, 1974

Yana qarang
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* Zaryadlar sistemasining uzoq masofalardagi kechikuvchi potensiallari
* Elektr dipolning elektromagnit maydoni va nurlanishi
* Nurlanish reaksiyasi