Zaryadlar sistemasi uchun
![{\displaystyle {\textbf {A}}={\frac {1}{c}}\sum {\frac {e_{i}v_{i}}{R_{i}}}={\frac {1}{c}}\sum {\frac {e_{i}v_{i}}{|{\textbf {r}}-{\textbf {r}}_{i}|}}\ ;(1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/387da8bea5aadabbb0bf7b402d9a5679bba39438)
bu yerda koordinatalar boshi sifatida sistemaning ichki nuqtalaridan biri olinib, oʻsha nuqtaga nisbatan kuzatish nuqtasining radius vektori r va
zaryadning radius vektori ri boʻlib, Ri esa kuzatish nuqtasining
zaryadga nisbatan radius-vektoridir.
Uzoq masofalardagi nuqtalar uchun
![{\displaystyle r_{i}<<r\ ;(2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8417b8336ac553223403a658d0b40f628dfd0796)
Endi (1) dagi masofaga teskari funksiyani
zaryad turgan nuqta koordinatalarining darajalari boʻyicha kuzatish nuqtasida Teylor qatoriga yoyib chiqaylik:
![{\displaystyle {\frac {1}{|{\textbf {r}}-{\textbf {r}}_{i}|}}={\frac {1}{r}}-x_{i}{\dfrac {\partial }{\partial x}}\left({\frac {1}{r}}\right)-y_{i}{\dfrac {\partial }{\partial y}}\left({\frac {1}{r}}\right)-z_{i}{\dfrac {\partial }{\partial z}}\left({\frac {1}{r}}\right)+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e952b168ab76c1dbd12eff9c68d0078a634d13b4)
bu yerda ikkinchi va yuqori darajali koordinatalar ishtirok etgan hadlarni yozib oʻtirmadik. Agar (2) ga muvofiq, bu hadlarn nazarga olinmasa,
![{\displaystyle {\frac {1}{|{\textbf {r}}-{\textbf {r}}_{i}|}}={\frac {1}{r}}-\left({\textbf {r}}_{i}\nabla {\frac {1}{r}}\right)={\frac {1}{r}}+{\textbf {r}}_{i}{\frac {\textbf {r}}{r^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc0cf83efa6cf27dc32ed8cf31eea1d83257461)
U vaqtda
![{\displaystyle {\textbf {A}}={\frac {1}{c{\textbf {r}}}}\sum e_{i}v_{i}+{\frac {1}{cr^{3}}}\sum e_{i}v_{i}\ ({\textbf {r}}_{i}{\textbf {r}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fea154d732be6536e96ab1d5a4589b561d70150f)
Maʼlumki,
![{\displaystyle \sum e_{i}v_{i}={\dfrac {d}{dt}}\sum e_{i}{\textbf {r}}_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/170799eeadd8c955e3ad6bd555bb6193c6bc5b57)
![{\displaystyle \sum e_{i}v_{i}\ ({\textbf {r}}_{i}{\textbf {r}})={\dfrac {d}{dt}}\sum e_{i}{\textbf {r}}_{i}\ ({\textbf {r}}_{i}{\textbf {r}})-\sum e_{i}{\textbf {r}}_{i}\ ({\textbf {v}}_{i}{\textbf {r}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae1785c699e60d08b6f45c6f1648693e03e0a885)
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,
,
. Soʻnggi tenglikdan koʻramizki,
![{\displaystyle 2\sum e_{i}{\textbf {v}}_{i}\ ({\textbf {r}}_{i}{\textbf {r}})=\sum e_{i}{\textbf {v}}_{i}\ ({\textbf {r}}_{i}{\textbf {r}})-\sum e_{i}{\textbf {r}}_{i}\ ({\textbf {v}}_{i}{\textbf {r}})=\sum e_{i}\left[{\textbf {r}}_{i}\left[{\textbf {v}}_{i}{\textbf {r}}_{i}\right]\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e35d0a3cc321b77cbdcaaa0aca57530d8ca74f0)
Shularni hisobga olsak,
![{\displaystyle {\textbf {A}}={\frac {1}{2er^{3}}}\sum e_{i}\left[{\textbf {r}}_{i}\left[{\textbf {v}}_{i}{\textbf {r}}_{i}\right]\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1cdcfb929099b0ba83534a56edb6c2a603ccdf0)
yoki
![{\displaystyle {\textbf {A}}=-\left[{\frac {\textbf {r}}{r^{3}}}\sum {\frac {e_{i}}{2c}}\left[{\textbf {r}}_{i}{\textbf {v}}_{i}\right]\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c814d234ffa7c9f762680cbbc290e6aaeec0c740)
boʻladi. Bu yerdagi yigʻindi vektorni M orqali belgilaylik, u holda
![{\displaystyle {\textbf {M}}=\sum {\frac {e_{i}}{2c}}\left[{\textbf {r}}_{i}{\textbf {v}}_{i}\right];\ \ \ (3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87d4189630ea13b86a139dd89185176b053a881e)
Shunday qilib,
![{\displaystyle {\textbf {A}}=-\left[{\frac {\textbf {r}}{r^{3}}}{\textbf {M}}\right];\ \ \ (4)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f955bfda1f8dcc7431223521afa7b1e971f3779)
yoki
![{\displaystyle {\textbf {A}}=\left[\nabla {\frac {1}{\textbf {r}}},{\textbf {M}}\right];\ \ \ (5)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b4f8ccf15be7e9cf1784dd09f84895c638beab6)
Nabla-operatorning simvolik vektor ekanligini va ikki qaytali vektor koʻpaytma xususiyatini esga olib, quyidagini yozamiz:
![{\displaystyle {\textbf {H}}={\textrm {rot}}\ {\textbf {A}}=\left[\nabla \left[\nabla {\frac {1}{r}},{\textbf {M}}\right]\right]=\left({\textbf {M}}\nabla \right)\nabla {\frac {1}{\textbf {r}}}-{\textbf {M}}\left(\nabla \ \nabla \right){\frac {1}{r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1093de935ba02c8716f7f67d0b4f8358047129e6)
Ammo
![{\displaystyle \left(\nabla \ \nabla \right){\frac {1}{r}}=\Delta {\frac {1}{r}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4989e0065cd6870374bb0c0216ec60b3d4166cfa)
![{\displaystyle \left({\textbf {M}}\nabla \right)\nabla {\frac {1}{\textbf {r}}}=-\left({\textbf {M}}\nabla \right){\frac {\textbf {r}}{r^{3}}}={\frac {\left({\textbf {M}}\nabla \right){\textbf {r}}}{r^{3}}}-{\textbf {r}}\left({\textbf {M}}\nabla {\frac {1}{r^{3}}}\right)=-{\frac {\textbf {M}}{r^{3}}}+{\frac {3\left({\textbf {Mr}}\right){\textbf {r}}}{r^{5}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/634a5e5caf09e4b70d455c5e4743de8d659c14df)
Demak,
![{\displaystyle {\textbf {H}}=-{\frac {\textbf {M}}{r^{3}}}+{\frac {3\left({\textbf {Mr}}\right){\textbf {r}}}{r^{5}}};\ \ \ (6)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a867aaedca613b78dcfafa5e07ebef9c161cb51e)
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.
Elektr dipol va magnit dipol maydonlari uzoq masofalardagina bir xil ifodalanadi va tasvirlanadi. Zaryadlar joylashgan sohada esa ular butunlay boshqacha. Haqiqatdan, bizga maʼlum boʻlgan
va
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[tahrir | manbasini tahrirlash]
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
va dastlabki
radius-vektorlari orasidagi bogʻlanish
boʻladi. Zaryadlar sistemasining yangi koordinatalar boshiga nisbatan magnit momenti
![{\displaystyle {\textbf {M}}'=\sum {\frac {e_{i}}{2c}}\left[{\textbf {r}}_{i}'{\textbf {v}}_{i}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f408c131199f519e28b1fbae835c49907861f0e)
yoki
![{\displaystyle {\textbf {M}}'=\sum {\frac {e_{i}}{2c}}\left[{\textbf {r}}_{i}{\textbf {v}}_{i}\right]+\sum {\frac {e_{i}}{2c}}\left[{\textbf {b}}{\textbf {v}}_{i}\right]=\sum {\frac {e_{i}}{2c}}\left[{\textbf {r}}_{i}{\textbf {v}}_{i}\right]+{\frac {1}{2c}}\left[{\textbf {b}}\sum e_{i}{\textbf {v}}_{i}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d24e649c6e93fe4236a4d7d426650df506b5e9a)
Yaʼni, (3) ga asosan,
![{\displaystyle {\textbf {M}}'={\textbf {M}}+{\frac {1}{2c}}\left[{\textbf {b}}\sum e_{i}{\textbf {v}}_{i}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3597025cd29900739b2214e8cd6ba6994bf5e488)
Ammo bilamizki, statsionar harakatdagi zaryadlar sistemasida yigʻindi
ning oʻrtacha qiymati nolga teng, yaʼni
.
- R.X.Mallin Klassik elektrodinamika, Oʻqituvchi, T., 1974