Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0000 360992 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0470 1209832 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0470 1209832 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1126 4102904 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1139 4130696 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1146 4148408 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1146 4165416 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1153 4173152 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1153 4173152 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1555 4402736 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1569 4420136 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1575 4441064 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.1661 4507680 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.1661 4507936 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182 Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; JCommentsACL has a deprecated constructor in /var/www/astarmathsandphysics/components/com_jcomments/classes/acl.php on line 17 Call Stack: 0.0000 360992 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0470 1209832 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0470 1209832 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1126 4102904 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1139 4130696 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1146 4148408 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1146 4165416 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1153 4173152 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1153 4173152 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1555 4402736 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1569 4420136 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1575 4441064 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2843 12301160 13. JEventDispatcher->trigger() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:199 0.2845 12301608 14. plgContentJComments->update() /var/www/astarmathsandphysics/libraries/joomla/event/dispatcher.php:160 0.2845 12301608 15. plgContentJComments->onContentAfterDisplay() /var/www/astarmathsandphysics/libraries/joomla/event/event.php:70 0.2847 12309856 16. plgContentJComments->onAfterDisplayContent() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:339 0.2849 12311592 17. JComments::show() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:282 0.2856 12344640 18. JCommentsFactory::getACL() /var/www/astarmathsandphysics/components/com_jcomments/jcomments.php:188 0.2856 12345040 19. spl_autoload_call() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274 0.2857 12345120 20. JLoader::load() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274

Flux Out of a Surface Contaioning Point Charges

Theorem
The flux out of any closed surface surrounding a system of charges  
\[ \{ q_i \}_i\]
  is  
\[\int \int S \mathbf{E} \cdot \mathbf{n} dS = \sum_i \frac{q_i}{\epsilon_0}\]

Proof
The potential at the point  
\[(x,y,z)\]
  on the surface due to the charge  
\[q_i\]
  is  
\[\phi_i = \frac{q_i}{4 \pi \epsilon_0 r_i}\]
  where  
\[r_i\]
  is the position vector of  
\[(x,y,z)\]
  relative to the charge.
The electric field at  
\[  on the surface due to the charge  
\[q_i\]
  is  
\[\mathbf{E_i} = - \mathbf{\nabla} \phi_i = - \frac{q_i}{4 \pi \epsilon_0} \mathbf{\nabla} (\frac{1}{r_i}) = \frac{q_i}{4 \pi \epsilon_0} \frac{\mathbf{r_i}}{r^3_i} \]

The electric field due to the collection of charges is
\[\mathbf{E} = \sum_i \mathbf{E_i} = \sum_i \frac{q_i}{4 \pi \epsilon_0} \frac{\mathbf{r_i}}{r^3_i} \]

The flux out of the surface  
\[S\]
  is
\[\int \int_S \mathbf{E} \cdot \mathbf{n} dS = \frac{1}{4 \pi \epsilon_0} \int \int_S \sum_i \frac{q_i \mathbf{r_i} \cdot \mathbf{n}}{r^3_i} dS\]

Now use Gauss's Theorem
to give
\[\frac{}{4 \pi \epsilon_0} \int \int_S \sum_i \frac{q_i \mathbf{r_i} \cdot \mathbf{n}}{r^3_i} dS=\frac{}{4 \pi \epsilon_0} \sum_i \frac{4 \pi q_i} = \sum_i \frac{q_i}{\epsilon_0} \]

Add comment

Security code
Refresh