Kechikuvchi potensiallar

Elektromagnit toʻlqinlar quyidagi differensial tenglamalarga boʻysunadi:

$\textrm{div}\textbf{A}+\frac{1}{c}\dfrac{\partial\varphi}{\partial t}=0;\ \ \ \ \ \ (1)$

$\Delta\textbf{A}-\frac{1}{c^{2}}\dfrac{\partial^{2}\textbf{A}}{\partial t^{2}}=-4\pi\frac{1}{c};\ \ \ \ \ \ (2)$

$\Delta\varphi-\frac{1}{c^{2}}\dfrac{\partial^{2}\varphi}{\partial t^{2}}=-4\pi\rho;\ \ \ \ \ \ \ (3)$

Koʻrinib turibdiki, xususiy hosilalardan iborat chiziqli differensial tenglamalar sistemasi bilan amal qilishga toʻgʻri keladi.

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

$\textrm{div}\textbf{A}+\frac{1}{c}\dfrac{\partial\varphi}{\partial t}=0;\ \ \ \ \ \ (4)$

$\Delta \textbf{A}-\frac{1}{c^{2}}\dfrac{\partial^{2}\textbf{A}}{\partial t^{2}}=0;\ \ \ \ \ \ (5)$

$\Delta\varphi-\frac{1}{c}\dfrac{\partial^{2}\varphi}{\partial t^{2}}=0;\ \ \ \ \ \ (6)$

$\delta e$ zaryadning $dv$ 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

$\frac{1}{R^{2}}\dfrac{\partial}{\partial R}\left(R^{2}\frac{\partial\varphi}{\partial R}\right)-\frac{1}{c^{2}}\dfrac{\partial^{2}\varphi}{\partial t^{2}}=0;\ \ \ \ \ \ (7)$

Quyidagi koʻrinishda yangi funksiya kiritamiz:

$\varphi(t)=\frac{\psi(R,t)}{R};\ \ \ \ \ \ (8)$

Koʻrinib turibdiki,

$\frac{\partial\varphi}{\partial R}=\frac{1}{R}\frac{\partial\psi}{\partial R}-\frac{\psi}{R^{2}},$

$R^{2}\frac{\partial\varphi}{\partial R}=R\frac{\partial\psi}{\partial R}-\psi,$

$\dfrac{\partial}{\partial R}\left(R^{2}\frac{\partial\varphi}{\partial R}\right)=R\frac{\partial^{2}\psi}{\partial R^{2}}+\frac{\partial\psi}{R}-\frac{\partial\psi}{R}=R\frac{\partial^{2}\psi}{\partial R^{2}},$

$\frac{1}{R^{2}}\dfrac{\partial}{\partial R}\left(R^{2}\frac{\partial\varphi}{\partial R}\right)=\frac{1}{R}\frac{\partial^{2}\psi}{\partial R^{2}}$

$\frac{1}{c^{2}}\dfrac{\partial^{2}\varphi}{\partial t^{2}}=\frac{1}{c^{2}}\frac{\partial^{2}\psi}{\partial t^{2}}$

Demak, (7) ga muvofiq

$\dfrac{\partial^{2}\psi}{\partial R^{2}}-\frac{1}{c}\dfrac{\partial^{2}\psi}{\partial t^{2}}=0;\ \ \ \ \ \ (9)$

boʻladi. Bu yassi elektromagnit to'lqin tenglamasidir. Shunday qilib,

$\psi=\psi_{1}\left(t+\frac{R}{c}\right)+\psi_{2}\left(t-\frac{R}{c}\right).\ \ \ \ \ \ \ (10)$

To'lqin tenglamaning xususiy yechimigina bizni qiziqtiradi. Shu sababli $\psi_{2}$ bilan cheklanamiz:

$\psi=\psi\left(t-\frac{R}{c}\right).\ \ \ \ \ \ (11)$

U holda, (8) ga muvofiq

$\varphi (t)=\frac{\psi\left(t-\frac{R}{c}\right)}{R}.\ \ \ \ \ \ (12)$

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

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

$\varphi(t)=\frac{\delta e(t)}{R}=\frac{\rho(t)dV}{R}$

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

$\varphi(t)=\frac{\rho\left(t-\frac{R}{c}\right)dV}{R}.\ \ \ \ \ \ (13)$

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

Zaryadlar sistemasining potensiali

Zaryadlar sistemasining potensialini aniqlash uchun (13) formulani sistemaning barcha zaryadlari joylashgan hajm boʻyicha integrallash lozim:

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

Xuddi shuningdek,

$\textbf{A}(t)=\frac{1}{c}\int \frac{\textbf{j}\left(t-\frac{R}{c}\right)}{R}\,dV;\ \ \ \ \ \ (15)$

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 $\textbf{r}^{\prime}$ va kuzatish nuqtasining radius-vektori $\textbf{r}$ bilan belgilanadi. Demak,

$\textbf{R}=\textbf{r}-\textbf{r}^{\prime};\ \ \ \ \ \ (16)$

boʻladi. Yangi koordinatalar boshiga nisbatan kechikuvchi potensiallar quyidagicha yoziladi:

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

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

Adabiyotlar

• R.X.Mallin, Klassik elektrodinamika, Toshkent, 1974

Yana qarang

• Zaryadlar sistemasining uzoq masofalardagi kechikuvchi potensiallari
• Elektr dipolning elektromagnit maydoni va nurlanishi
• Nurlanish reaksiyasi