Уравнения электромагнитной волны

Электромагнитными волнами называют распространяющееся в пространстве переменное электромагнитное поле.

Электромагнитные волны являются поперечными: векторы \overline{E} и \overline{H} напряженностей электрического и магнитного полей волны взаимно перпендикулярны и лежат в плоскости, и лежат в плоскости перпендикулярной вектору \overline{v} скорости распространения волны. Векторы \overline{v},\ \overline{E} и \overline{H} образуют правовинтовую систему.

Уравнения электромагнитной волны

Связь между векторами \overline{E} и \overline{H} в электромагнитной волне, распространяющейся в непроводящей среде, определяется уравнениями Максвелла, в которых \rho (плотность заряда) и \overline{j} (вектор плотности тока) полагают равными нулю:

   <pre class=*** QuickLaTeX cannot compile formula: \[rot\overline{E}=-\frac{\partial \overline{B}}{\partial t},\ div\overline{D}=0\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»201″ height=»42″ />

   <pre class=*** QuickLaTeX cannot compile formula: \[rot\overline{H}=\frac{\partial \overline{D}}{\partial t},\ div\overline{B}=0\ \qquad (1)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»255″ height=»42″ />

где \overline{E}-напряженность электрического поля, \overline{B}-вектор магнитной индукции, \overline{H}-вектор магнитной напряженности, \overline{D}— вектор электрического смещения.

Таким образом, система уравнений (1) есть уравнения электромагнитной волны в непроводящей среде.

В случае однородной, изотропной, непроводящей среды, не обладающей ферромагнитными или сегнетоэлектрическими свойствами уравнения электромагнитной волны будут иметь вид:

   <pre class=*** QuickLaTeX cannot compile formula: \[\overline{D}=\varepsilon {\varepsilon}_0\overline{E},\ \overline{B}=\mu {\mu}_0\overline{H}\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»186″ height=»20″ />

   <pre class=*** QuickLaTeX cannot compile formula: \[rot\overline{E}=-\mu {\mu}_0\frac{\partial \overline{H}}{\partial t},\ div\overline{E}=0\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»232″ height=»42″ />

   <pre class=*** QuickLaTeX cannot compile formula: \[rot\overline{H}=\varepsilon {\varepsilon}_0\frac{\partial \overline{E}}{\partial t},\ div\overline{H}=0\ \qquad (2)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»281″ height=»42″ />

где {\varepsilon}_0,\ {\mu}_0 – электрическая и магнитная постоянные, \varepsilon, \mu – относительные электрическая и магнитная проницаемость среды.

Векторы \overline{E} и \overline{H} поля электромагнитной волны можно выразить через скалярный \varphi и векторный \overline{A} потенциалы. Тогда уравнения электромагнитной волны будут иметь вид:

   <pre class=*** QuickLaTeX cannot compile formula: \[\overline{E}=-\frac{\partial \overline{A}}{\partial t}-grad\varphi ,\ \overline{H}=\frac{1}{\mu {\mu}_0}rot\overline{A}\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»290″ height=»45″ />

   <pre class=*** QuickLaTeX cannot compile formula: \[\Delta \varphi =\frac{\varepsilon \mu}{c^2}\frac{{\partial}^2\varphi}{\partial t^2},\ \Delta \overline{A}=\frac{\varepsilon \mu}{c^2}\frac{{\partial}^2\overline{A}}{\partial t^2}\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»242″ height=»42″ />

   <pre class=*** QuickLaTeX cannot compile formula: \[\Delta \overline{E}=\frac{\varepsilon \mu}{c^2}\frac{{\partial}^2\overline{E}}{\partial t^2},\ \Delta \overline{H}=\frac{\varepsilon \mu}{c^2}\frac{{\partial}^2\overline{H}}{\partial t^2}, \qquad (3)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»323″ height=»42″ />

где \Delta— оператор Лапласа, c=\frac{1}{\sqrt{{\varepsilon}_0{\mu}_0}}=3\cdot {10}^8\frac{m}{c},\ div\ \overline{A}=0.

Каждая из проекций векторов \overline{A},\ \overline{E},\ \overline{H} на оси прямоугольной декартовой системы координат и \varphi удовлетворяют волновому уравнению(4):

   <pre class=*** QuickLaTeX cannot compile formula: \[\Delta S_i=\frac{1}{v^2}\frac{{\partial}^2s_i}{\partial t^2}\ \left(i=1,2,\dots ,10\right),\ \qquad (4)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»339″ height=»42″ />

где v=\frac{c}{\sqrt{\varepsilon \mu}} – фазовая скорость электромагнитной волны, s_1=\varphi,\ s_2=A_x,\ s_3=A_y,\ \dots ,\ s_{10}=H_z. В вакууме (\varepsilon = \mu =1).\ v=c. Для всех сред кроме ферромагнитных, \mu \approx 1 и v=\frac{c}{\sqrt{\varepsilon}}.

Определение и уравнение плоской электромагнитной волны

ОПРЕДЕЛЕНИЕ

Электромагнитную волну называют плоской, если векторы \overline{E} и \overline{H} зависят только от времени и одной декартовой координаты.

Для плоской электромагнитной волны, распространяющейся вдоль положительного направления оси Ox правовинтовой системы координат, уравнения электромагнитной волны запишутся в следующем виде:

   <pre class=*** QuickLaTeX cannot compile formula: \[E_x=H_x=0\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»107″ height=»15″ />

   <pre class=*** QuickLaTeX cannot compile formula: \[\frac{\partial E_y}{\partial x}=-\mu {\mu}_0\frac{\partial H_z}{\partial t},\ \frac{\partial E_z}{\partial x}=-\mu {\mu}_0\frac{\partial H_y}{\partial t}\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»306″ height=»41″ />

   <pre class=*** QuickLaTeX cannot compile formula: \[\frac{\partial H_y}{\partial x}=-\varepsilon {\varepsilon}_0\frac{\partial E_z}{\partial t},\ \frac{\partial H_z}{\partial x}=-\varepsilon {\varepsilon}_0\frac{\partial E_y}{\partial t}\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»295″ height=»41″ />

   <pre class=*** QuickLaTeX cannot compile formula: \[H_z=\sqrt{\frac{\varepsilon {\varepsilon}_0}{\mu {\mu}_0}}E_y,\ H_y=-\sqrt{\frac{\varepsilon {\varepsilon}_0}{\mu {\mu}_0}}E_z\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»278″ height=»46″ />

   <pre class=*** QuickLaTeX cannot compile formula: \[\overline{E}=\sqrt{\frac{\mu {\mu}_0}{\varepsilon {\varepsilon}_0}}\ \overline{H}\cdot \overline{n}=\frac{c}{\sqrt{\varepsilon \mu}} rot \overline{A} \times \overline{n},\qquad (5)\ \] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»348″ height=»46″ />

где \overline{n} – единичный вектор, проведенный в направлении распространении волны. Мы видим, что плоская электромагнитная волна может быть полностью определена с помощью одного лишь векторного потенциала \overline{A}. В вакууме:

   <pre class=*** QuickLaTeX cannot compile formula: \[H_y=-\sqrt{\frac{{\varepsilon}_0}{{\mu}_0}}E_z,\ H_z=\sqrt{\frac{{\varepsilon}_0}{{\mu}_0}}E_y\ \qquad (6)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»323″ height=»46″ />

   <pre class=*** QuickLaTeX cannot compile formula: \[\sqrt{{\varepsilon}_0}E=\sqrt{{\mu}_0}\ H\ \qquad (7)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»196″ height=»19″ />

Определение и уравнение монохроматической электромагнитной волны

ОПРЕДЕЛЕНИЕ

Электромагнитную волну называют монохроматической, если компоненты векторов \overline{E} и \overline{H} электромагнитного поля волны совершают гармонические колебания одинаковой частоты, называемой частотой волны. Произвольная немонохроматическая волна может быть представлена в виде совокупности монохроматических волн.

Напряженности электрического и магнитного полей плоской монохроматической волны часто представляют в виде:

   <pre class=*** QuickLaTeX cannot compile formula: \[\overline{E}=\overline{E_0}{ \cos \left(\overline{k}\overline{r}-wt\right)},\ \overline{H}=\overline{H_0}\cdot { \cos \left(\overline{k}\overline{r}-wt\right)} \qquad (8)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»433″ height=»24″ />

где k=w/v – волновое число.

Очевидно, что и комплексные функции эквивалентны выражению (8)

   <pre class=*** QuickLaTeX cannot compile formula: \[\overline{E}\left(r,t\right)=\overline{E_0}{\exp \left(i\overline{k}\overline{r}-iwt\right)},\ \overline{H}\left(r,t\right)=\overline{H_0}{exp \left(i\overline{k}\overline{r}-iwt\right)} \qquad (9)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»535″ height=»23″ />

Здесь \overline{k} — волновой вектор, а его модуль равен величине k = \omega /r.

Электрическое и магнитное поля плоской волны направлены перпендикулярно друг-другу.

Примеры решения задач

ПРИМЕР 1

Задание Найти средний вектор Пойнтинга у плоской электромагнитной волны \overline{E}={\overline{E}}_m \cos (wt-\overline{k}\overline{r}), если волна распространяется в вакууме.
Решение По определению вектор Пойнтинга:

   <pre class=*** QuickLaTeX cannot compile formula: \[\overline{S}=\left[\overline{E}\times \overline{H}\right]\qquad (1.1)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»185″ height=»23″ />

Запишем средний вектор Пойнтинга, т.к \overline{E}\bot \overline{H}:

   <pre class=*** QuickLaTeX cannot compile formula: \[<S>=\frac{\left|\overline{S}\right|}{2}=\frac{\left|E\right|\left|H\right|}{2}\qquad (1.2)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»260″ height=»43″ />

Для плоской электромагнитной волны E_x=H_x=0, тогда положим:

   <pre class=*** QuickLaTeX cannot compile formula: \[E_y=E \cos \left(wt-kx+{\alpha}_1\right),\qquad (1.3)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»302″ height=»20″ />

   <pre class=*** QuickLaTeX cannot compile formula: \[H_z=H \cos \left(wt-kx+{\alpha}_2\right)\qquad (1.4)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»297″ height=»18″ />

Запишем уравнения полоской электромагнитной волны:

   <pre class=*** QuickLaTeX cannot compile formula: \[\frac{dE_y}{dx}=-\mu {\mu}_0\frac{dH_z}{dt},\qquad (1.5)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»225″ height=»39″ />

   <pre class=*** QuickLaTeX cannot compile formula: \[\frac{dH_z}{dx}=-\varepsilon {\varepsilon}_0\frac{dE_y}{dt}\qquad (1.6)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»211″ height=»39″ />

Продифференцируем (1.3) по x:

   <pre class=*** QuickLaTeX cannot compile formula: \[\frac{dE_y}{dx}=\frac{d}{dx}\left(E \cos \left(wt-kx+б_1\right)\right)=-Ek \sin \left(-wt+kx-{\alpha}_1\right)\qquad (1.7)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»567″ height=»39″ />

Продифференцируем (1.4) по t, получим:

   <pre class=*** QuickLaTeX cannot compile formula: \[\frac{dH_z}{dt}=Hw \sin (-wt+kx-{\alpha}_2)\qquad (1.8)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»329″ height=»39″ />

Подставим (1.7) и (1.8) в (1.5), получим:

   <pre class=*** QuickLaTeX cannot compile formula: \[-Ek \sin \left(-wt+kx-{\alpha}_1\right)=-\mu \mu_0Hw \sin ( -wt+kx-{\alpha}_2)\qquad (1.9)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»544″ height=»18″ />

   <pre class=*** QuickLaTeX cannot compile formula: \[H=\frac{kE}{{\mu}_0\mu w} \qquad (1.10)\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»175″ height=»43″ />

Подставим (1.10) в (1.1), затем в (1.2), получим:

   <pre class=*** QuickLaTeX cannot compile formula: \[<S>=\frac{1}{2}E^2\frac{k}{{\mu}_0\mu w}\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»154″ height=»43″ />

Так как волна распространяется в вакууме, то запишем:

   <pre class=*** QuickLaTeX cannot compile formula: \[\mu =1\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»44″ height=»16″ />

   <pre class=*** QuickLaTeX cannot compile formula: \[{\mu}_0{\varepsilon}_0=\frac{1}{c^2},\ {\mu}_0=\frac{1}{c^2{\varepsilon}_0}\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»177″ height=»41″ />

   <pre class=*** QuickLaTeX cannot compile formula: \[<S>=\frac{1}{2}E^2\frac{k\cdot c^2{\varepsilon}_0}{w}\] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»167″ height=»41″ />
Ответ Средний вектор Пойнтинга у плоской электромагнитной волны:

   <pre class=*** QuickLaTeX cannot compile formula: \[ <S> =\frac{1}{2}E^2\frac{k\cdot c^2{\varepsilon}_0}{w} \] *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

» width=»167″ height=»41″ />