Различия

Показаны различия между двумя версиями страницы.

--- optics:res4.4 [2020/05/14 11:05] – root
+++ optics:res4.4 [2020/05/14 11:17] (текущий) – root
@@ Строка 34: / Строка 34: @@
 $$\frac{1}{cr}\frac{\partial}{\partial t'}\vec{p}'(t')=\frac{i\omega}{cr}\vec{p}_{0}e^{i\omega t'}=\frac{i\omega}{cr}\vec{p}_{0}e^{i\omega\left(t-\frac{R}{c}\right)}$$
 $$\vec{B}=\text{rot }\vec{A}.$$
-Далее будем считать $R\approx r$, и в сферических координатах $$A_{\theta}=-A\sin\theta,\,A_{r}=A\cos\theta$$
+Далее будем считать $R\approx r$, и в сферических координатах $$A_{\theta}=-A\sin\theta,\,A_{r}=A\cos\theta .$$
-найдём поле. Так как
+Найдём поле. Так как
 $$\text{rot}{}_{r}\vec{A}=\frac{1}{r\sin\theta}\left(\frac{\partial\left(\sin\theta A_{\alpha}\right)}{\partial\theta}-\frac{\partial A_{\theta}}{\partial\alpha}\right),
 $$
@@ Строка 44: / Строка 44: @@
 \text{rot}{}_{\alpha}\vec{A}=\frac{1}{r}\frac{\partial\left(rA_{\theta}\right)}{\partial r}-\frac{1}{r}\frac{\partial A_{r}}{\partial\theta},$$
 то ненулевая компонента
-$$B_{\alpha}=\text{rot}{}_{\alpha}\vec{A}=\frac{1}{r}\frac{i\omega}{c}p_{0}\frac{\partial\left(e^{i\omega\left(t-\frac{r}{c}\right)}\right)\sin\theta}{\partial r}-\frac{i\omega}{cr^{2}}p_{0}e^{i\omega\left(t-\frac{r}{c}\right)}\frac{\partial\cos\theta}{\partial\theta}=$$
+$$B_{\alpha}=\text{rot}{}_{\alpha}\vec{A}=\frac{1}{r}\frac{i\omega}{c}p_{0}\frac{\partial\left(e^{i\omega\left(t-\frac{r}{c}\right)}\right)\sin\theta}{\partial r}+\frac{i\omega}{cr^{2}}p_{0}e^{i\omega\left(t-\frac{r}{c}\right)}\frac{\partial\cos\theta}{\partial\theta}=$$
-$$\frac{1}{r}\frac{\omega^{2}}{c^{2}}p_{0}e^{i\omega\left(t-\frac{r}{c}\right)}\sin\theta+\frac{i\omega}{cr^{2}}p_{0}e^{i\omega\left(t-\frac{r}{c}\right)}\sin\theta=
+$$\frac{1}{r}\frac{\omega^{2}}{c^{2}}p_{0}e^{i\omega\left(t-\frac{r}{c}\right)}\sin\theta-\frac{i\omega}{cr^{2}}p_{0}e^{i\omega\left(t-\frac{r}{c}\right)}\sin\theta=
 $$
-$$\left(\frac{\omega}{c}+\frac{i}{r}\right)\frac{\omega}{cr}p_{0}e^{i\omega\left(t-\frac{r}{c}\right)}\sin\theta.$$
+$$\left(\frac{\omega}{c}-\frac{i}{r}\right)\frac{\omega}{cr}p_{0}e^{i\omega\left(t-\frac{r}{c}\right)}\sin\theta.$$
 Электрическое поле в силу $$\frac{1}{c}\frac{\partial\vec{E}}{\partial t}=\text{rot }\vec{H}$$
 будем искать в виде $$\vec{E}=\vec{E}_{0}e^{i\omega\left(t-\frac{r}{c}\right)},$$
@@ Строка 53: / Строка 53: @@
 $$\frac{i\omega}{c}E_{r}=\text{rot}{}_{r}\vec{B}=\frac{1}{r\sin\theta}\left(\frac{\partial\left(\sin\theta B_{\alpha}\right)}{\partial\theta}-\frac{\partial B_{\theta}}{\partial\alpha}\right)=$$
 $$
-\frac{1}{r\sin\theta}\left(\frac{\omega}{c}+\frac{i}{r}\right)\frac{\omega}{cr}p_{0}e^{i\omega\left(t-\frac{r}{c}\right)}\frac{\partial\left(\sin^{2}\theta\right)}{\partial\theta}=$$
+\frac{1}{r\sin\theta}\left(\frac{\omega}{c}-\frac{i}{r}\right)\frac{\omega}{cr}p_{0}e^{i\omega\left(t-\frac{r}{c}\right)}\frac{\partial\left(\sin^{2}\theta\right)}{\partial\theta}=$$
 $$
-\left(\frac{\omega}{c}+\frac{i}{r}\right)\frac{\omega}{cr^{2}}p_{0}e^{i\omega\left(t-\frac{r}{c}\right)}\cos\theta.$$
+\left(\frac{\omega}{c}-\frac{i}{r}\right)\frac{\omega}{cr^{2}}p_{0}e^{i\omega\left(t-\frac{r}{c}\right)}\cos\theta.$$
 $$\frac{i\omega}{c}E_{\theta}=\text{rot}{}_{\theta}\vec{B}=\frac{1}{r\sin\theta}\frac{\partial B_{r}}{\partial\alpha}-\frac{1}{r}\frac{\partial\left(rB_{\alpha}\right)}{\partial r}=$$
 $$
--\frac{\frac{\omega}{c}p_{0}\sin\theta}{r}\frac{\partial\left(\left(\frac{\omega}{c}+\frac{i}{r}\right)e^{i\omega\left(t-\frac{r}{c}\right)}\right)}{\partial r}=$$
+-\frac{\frac{\omega}{c}p_{0}\sin\theta}{r}\frac{\partial\left(\left(\frac{\omega}{c}-\frac{i}{r}\right)e^{i\omega\left(t-\frac{r}{c}\right)}\right)}{\partial r}=$$
-$$\frac{i\frac{\omega}{c}p_{0}\sin\theta}{r^{3}}e^{i\omega\left(t-\frac{r}{c}\right)}+\frac{i\frac{\omega^{2}}{c^{2}}p_{0}\sin\theta}{r}\left(\frac{\omega}{c}+\frac{i}{r}\right)e^{i\omega\left(t-\frac{r}{c}\right)}=$$
+$$\frac{i\frac{\omega}{c}p_{0}\sin\theta}{r^{3}}e^{i\omega\left(t-\frac{r}{c}\right)}+\frac{i\frac{\omega^{2}}{c^{2}}p_{0}\sin\theta}{r}\left(\frac{\omega}{c}-\frac{i}{r}\right)e^{i\omega\left(t-\frac{r}{c}\right)}=$$
 $$
-\left(\frac{1}{r^{2}}+\frac{\omega^{2}}{c^{2}}+\frac{i\omega}{rc}\right)i\frac{\omega}{rc}p_{0}\sin\theta e^{i\omega\left(t-\frac{r}{c}\right)}.$$
+\left(\frac{1}{r^{2}}+\frac{\omega^{2}}{c^{2}}-\frac{i\omega}{rc}\right)i\frac{\omega}{rc}p_{0}\sin\theta e^{i\omega\left(t-\frac{r}{c}\right)}.$$
 Тогда
-$$E_{r}=2\left(\frac{1}{r}-\frac{i\omega}{c}\right)\frac{p_{0}e^{i\omega\left(t-\frac{r}{c}\right)}\cos\theta}{r^{2}},$$
+$$E_{r}=-2\left(\frac{1}{r}+\frac{i\omega}{c}\right)\frac{p_{0}e^{i\omega\left(t-\frac{r}{c}\right)}\cos\theta}{r^{2}},$$
-$$E_{\theta}=\left(\frac{1}{r^{2}}+\frac{\omega^{2}}{c^{2}}+\frac{i\omega}{rc}\right)\frac{p_{0}\sin\theta e^{i\omega\left(t-\frac{r}{c}\right)}}{r}.$$
+$$E_{\theta}=\left(\frac{1}{r^{2}}+\frac{\omega^{2}}{c^{2}}-\frac{i\omega}{rc}\right)\frac{p_{0}\sin\theta e^{i\omega\left(t-\frac{r}{c}\right)}}{r}.$$