Zaryadlar sistemasining uzoq masofalardagi magnit maydoni

Magnit maydonning vektor potensialini aniqlash

Zaryadlar sistemasi uchun

$\mathbf{\text{A}}=\frac{1}{c}\sum \frac{e_i{v}_i}{R_i}=\frac{1}{c}\sum \frac{e_i{v}_i}{|\mathbf{\text{r}}-{\mathbf{\text{r}}}_i|};(1)$

bu yerda koordinatalar boshi sifatida sistemaning ichki nuqtalaridan biri olinib, oʻsha nuqtaga nisbatan kuzatish nuqtasining radius vektori r va $e_i$ zaryadning radius vektori r_i boʻlib, R_i esa kuzatish nuqtasining $e_i$ zaryadga nisbatan radius-vektoridir.

Uzoq masofalardagi nuqtalar uchun

$r_i<<r;(2)$

Endi (1) dagi masofaga teskari funksiyani $e_i$ zaryad turgan nuqta koordinatalarining darajalari boʻyicha kuzatish nuqtasida Teylor qatoriga yoyib chiqaylik:

$\frac{1}{|\mathbf{\text{r}}-{\mathbf{\text{r}}}_i|}=\frac{1}{r}-x_i\frac{\partial }{\partial x}\left(\frac{1}{r}\right)-y_i\frac{\partial }{\partial y}\left(\frac{1}{r}\right)-z_i\frac{\partial }{\partial z}\left(\frac{1}{r}\right)+\dots$

bu yerda ikkinchi va yuqori darajali koordinatalar ishtirok etgan hadlarni yozib oʻtirmadik. Agar (2) ga muvofiq, bu hadlarn nazarga olinmasa,

$\frac{1}{|\mathbf{\text{r}}-{\mathbf{\text{r}}}_i|}=\frac{1}{r}-\left({\mathbf{\text{r}}}_i\mathrm{\nabla }\frac{1}{r}\right)=\frac{1}{r}+{\mathbf{\text{r}}}_i\frac{\mathbf{\text{r}}}{r^3}$

U vaqtda

$\mathbf{\text{A}}=\frac{1}{c\mathbf{\text{r}}}\sum e_i{v}_i+\frac{1}{c{r}^3}\sum e_i{v}_i({\mathbf{\text{r}}}_i\mathbf{\text{r}})$

Maʼlumki,

$\sum e_i{v}_i=\frac{d}{dt}\sum e_i{\mathbf{\text{r}}}_i$

$\sum e_i{v}_i({\mathbf{\text{r}}}_i\mathbf{\text{r}})=\frac{d}{dt}\sum e_i{\mathbf{\text{r}}}_i({\mathbf{\text{r}}}_i\mathbf{\text{r}})-\sum e_i{\mathbf{\text{r}}}_i({\mathbf{\text{v}}}_i\mathbf{\text{r}})$

boʻladi (kuzatish nuqtasining radius-vektori r ni oʻzgarmas hisoblanadi). Statsionar harakatdagi zaryadlar sistemasiga tegishli funksiyalarning oʻrtacha qiymatlaridan vaqt boʻyicha olingan hosilalar nolga teng boʻlishi bizga maʼlum. Demak, $\sum e_i{\mathbf{\text{v}}}_i=0$, $\sum e_i{v}_i({\mathbf{\text{r}}}_i\mathbf{\text{r}})=-\sum e_i{\mathbf{\text{r}}}_i({\mathbf{\text{v}}}_i\mathbf{\text{r}})$. Soʻnggi tenglikdan koʻramizki,

$2\sum e_i{\mathbf{\text{v}}}_i({\mathbf{\text{r}}}_i\mathbf{\text{r}})=\sum e_i{\mathbf{\text{v}}}_i({\mathbf{\text{r}}}_i\mathbf{\text{r}})-\sum e_i{\mathbf{\text{r}}}_i({\mathbf{\text{v}}}_i\mathbf{\text{r}})=\sum e_i\left[{\mathbf{\text{r}}}_i\left[{\mathbf{\text{v}}}_i{\mathbf{\text{r}}}_i\right]\right]$

Shularni hisobga olsak,

$\mathbf{\text{A}}=\frac{1}{2e{r}^3}\sum e_i\left[{\mathbf{\text{r}}}_i\left[{\mathbf{\text{v}}}_i{\mathbf{\text{r}}}_i\right]\right]$

yoki

$\mathbf{\text{A}}=-[\frac{\mathbf{\text{r}}}{r^3}\sum \frac{e_i}{2c}\left[{\mathbf{\text{r}}}_i{\mathbf{\text{v}}}_i\right]]$

boʻladi. Bu yerdagi yigʻindi vektorni M orqali belgilaylik, u holda

$\mathbf{\text{M}}=\sum \frac{e_i}{2c}\left[{\mathbf{\text{r}}}_i{\mathbf{\text{v}}}_i\right];(3)$

Shunday qilib,

$\mathbf{\text{A}}=-\left[\frac{\mathbf{\text{r}}}{r^3}\mathbf{\text{M}}\right];(4)$

yoki

$\mathbf{\text{A}}=[\mathrm{\nabla }\frac{1}{\mathbf{\text{r}}},\mathbf{\text{M}}];(5)$

Magnit maydon kuchlanganligini aniqlash

Nabla-operatorning simvolik vektor ekanligini va ikki qaytali vektor koʻpaytma xususiyatini esga olib, quyidagini yozamiz:

$\mathbf{\text{H}}=\text{rot}\mathbf{\text{A}}=\left[\mathrm{\nabla }[\mathrm{\nabla }\frac{1}{r},\mathbf{\text{M}}]\right]=\left(\mathbf{\text{M}}\mathrm{\nabla }\right)\mathrm{\nabla }\frac{1}{\mathbf{\text{r}}}-\mathbf{\text{M}}\left(\mathrm{\nabla }\mathrm{\nabla }\right)\frac{1}{r}$

Ammo

$\left(\mathrm{\nabla }\mathrm{\nabla }\right)\frac{1}{r}=\mathrm{\Delta }\frac{1}{r}=0$

$\left(\mathbf{\text{M}}\mathrm{\nabla }\right)\mathrm{\nabla }\frac{1}{\mathbf{\text{r}}}=-\left(\mathbf{\text{M}}\mathrm{\nabla }\right)\frac{\mathbf{\text{r}}}{r^3}=\frac{\left(\mathbf{\text{M}}\mathrm{\nabla }\right)\mathbf{\text{r}}}{r^3}-\mathbf{\text{r}}\left(\mathbf{\text{M}}\mathrm{\nabla }\frac{1}{r^3}\right)=-\frac{\mathbf{\text{M}}}{r^3}+\frac{3\left(\mathbf{\text{Mr}}\right)\mathbf{\text{r}}}{r^5}$

Demak,

$\mathbf{\text{H}}=-\frac{\mathbf{\text{M}}}{r^3}+\frac{3\left(\mathbf{\text{Mr}}\right)\mathbf{\text{r}}}{r^5};(6)$

Bu formula bizga elektr dipolning elektr maydon kuchlanganligini eslatadi, faqat elektr moment p yangi M vektor bilan almashtirilgan. Shu qiyosga binoan M vektor magnit moment va unga ega zaryadlar sistemasi magnit dipol deyiladi. Shunday qilib, harakatlanuvchi zaryadlar sistemasining uzoq masofalardagi magnit maydoni shu sistemaning magnit dipol sifatida yaratgan magnit maydonidan iboratdir.

Bizga maʼlumki, zaryadlar sistemasining uzoq masofalarda yaratgan maydoni umuman, turli tartibdagi multipollar sistemasining yaratgan maydoni deb qaraladi. Yuqorida keltirilgan mulohazalar va hisoblashlardan ravshanki, harakatlanuvchi zaryadlar sistemasining uzoq masofalarda yaratgan magnit maydoni umuman, turli tartibdagi mos olingan magnit multipollar sistemasining yaratganmagnit maydoni kabidir. Birinchi tartibli magnit multipol magnit dipoldir, ikkinchi tartiblisi magnit kvadrupol va hokazo.

Eslatma

Elektr dipol va magnit dipol maydonlari uzoq masofalardagina bir xil ifodalanadi va tasvirlanadi. Zaryadlar joylashgan sohada esa ular butunlay boshqacha. Haqiqatdan, bizga maʼlum boʻlgan $\text{div}\mathbf{\text{E}}=4\pi \rho$ va $\text{div}\mathbf{\text{H}}=0$ ifodalardan koʻramizki, elektr dipolning elektr maydon chiziqlari musbat zaryaddan boshlanib, manfy zaryadda tugasa, magnit dipolning magnit maydoni yopiq chiziqlargina hosil qiladi.

Magnit momentining koordinatalar boshiga bogʻliqligi

Neytral sistemaning elektr momenti koordinatalar boshining tanlanishiga bogʻliq emasligini yaxshi bilamiz. Xuddi shuningdek, zaryadlar sistemasining magnit momenti koordinatalar boshining tanlanishiga bogʻliq emas. Haqiqatdan, yangi koordinatalar boshining dastlabki koordinatalar boshiga nisbatan siljish vektorini b orqali belgilasak, zaryadning yangi ${\mathbf{\text{r}}}_{i^{\prime }}$ va dastlabki ${\mathbf{\text{r}}}_i$ radius-vektorlari orasidagi bogʻlanish ${\mathbf{\text{r}}}_{i^{\prime }}={\mathbf{\text{r}}}_i+\mathbf{\text{b}}$ boʻladi. Zaryadlar sistemasining yangi koordinatalar boshiga nisbatan magnit momenti

${\mathbf{\text{M}}}^{\prime }=\sum \frac{e_i}{2c}\left[{\mathbf{\text{r}}}_{i^{\prime }}{\mathbf{\text{v}}}_i\right]$

yoki

${\mathbf{\text{M}}}^{\prime }=\sum \frac{e_i}{2c}\left[{\mathbf{\text{r}}}_i{\mathbf{\text{v}}}_i\right]+\sum \frac{e_i}{2c}\left[\mathbf{\text{b}}{\mathbf{\text{v}}}_i\right]=\sum \frac{e_i}{2c}\left[{\mathbf{\text{r}}}_i{\mathbf{\text{v}}}_i\right]+\frac{1}{2c}[\mathbf{\text{b}}\sum e_i{\mathbf{\text{v}}}_i]$

Yaʼni, (3) ga asosan,

${\mathbf{\text{M}}}^{\prime }=\mathbf{\text{M}}+\frac{1}{2c}[\mathbf{\text{b}}\sum e_i{\mathbf{\text{v}}}_i]$

Ammo bilamizki, statsionar harakatdagi zaryadlar sistemasida yigʻindi $e_i{\mathbf{\text{v}}}_i$ ning oʻrtacha qiymati nolga teng, yaʼni ${\mathbf{\text{M}}}^{\prime }=\mathbf{\text{M}}$.

Yana qarang

• Magnit maydonning toklar sistemasiga taʼsiri
• Larmor teoremasi
• Yassi elektromagnit toʻlqin
• Ossilyatorning radiatsion soʻnishi

Manba

• R.X.Mallin Klassik elektrodinamika, Oʻqituvchi, T., 1974