La Corrigé d'Examen 3 d'électrostatique
A/ σ > 0
1) * ![](https://3.bp.blogspot.com/-xGAmEaO5cvU/VzIOSMYOkwI/AAAAAAAAKiY/8tNO398T40sCZEYO83Eoy8Nlrgw0nNP0wCLcB/s1600/image002.gif)
![](https://3.bp.blogspot.com/-xGAmEaO5cvU/VzIOSMYOkwI/AAAAAAAAKiY/8tNO398T40sCZEYO83Eoy8Nlrgw0nNP0wCLcB/s1600/image002.gif)
![](https://1.bp.blogspot.com/-emHUEAGKdEc/VzIOSEdiJyI/AAAAAAAAKic/PNnuJ2cMRi0Xz5lUyG2TvzbovNcu2FyWQCLcB/s1600/image003.gif)
D’où, le champ est radial :
![](https://2.bp.blogspot.com/-BAK34vIhYCs/VzIOSu1VGyI/AAAAAAAAKig/9WpVhOqBFkkIynEvitduVKrpDqXhmkYRQCLcB/s1600/image004.gif)
2) a)
* Σ cylindre de hauteur h et d’axe z’z et passant par M.
* (S) présente une symétrie cylindrique et
est radial.
![](https://3.bp.blogspot.com/-y_qqnszc8oo/VzIOSkGBZ9I/AAAAAAAAKik/bapX8A4zv-AdndE6a7VufGJ8-soyfoRVwCLcB/s1600/image005.gif)
![](https://1.bp.blogspot.com/-KLW47NkAFgs/VzIOSzcq8OI/AAAAAAAAKio/g4wZGTOuGjw3gJEwSZuSToeHs8DdTJVWwCLcB/s1600/image006.gif)
3) a)
![](https://2.bp.blogspot.com/-qxBEtAupWs0/VzIOS7ZJfhI/AAAAAAAAKis/YJfHDeEMf-szIJSfCI82BIQhS6cH5wt_ACLcB/s1600/image007.gif)
b) E est discontinu en r = R et la valeur de la discontinuité vaut ![](https://3.bp.blogspot.com/-K6H30FBnkkA/VzIOS_s5uZI/AAAAAAAAKiw/xTS2xjoocjIS7SpsL80v1YzKtEbFxojOwCLcB/s1600/image008.gif)
![](https://3.bp.blogspot.com/-K6H30FBnkkA/VzIOS_s5uZI/AAAAAAAAKiw/xTS2xjoocjIS7SpsL80v1YzKtEbFxojOwCLcB/s1600/image008.gif)
4) ![](https://4.bp.blogspot.com/-nhxAIFqqQGE/VzIOTbOfR5I/AAAAAAAAKi0/Rtw0QRaCqdkEC9rYwFeIrs9jiv8kRm76wCLcB/s1600/image009.gif)
![](https://4.bp.blogspot.com/-nhxAIFqqQGE/VzIOTbOfR5I/AAAAAAAAKi0/Rtw0QRaCqdkEC9rYwFeIrs9jiv8kRm76wCLcB/s1600/image009.gif)
![](https://4.bp.blogspot.com/-ENMzUAI41uQ/VzIOTY6PHSI/AAAAAAAAKi4/jGfCR95qwGAm6lr6OkPV51trGaaVdwjkQCLcB/s1600/image010.gif)
B est déterminé par continuité du potentiel en r = R :
![](https://1.bp.blogspot.com/-zHspCXFP1Ls/VzIOTV35bbI/AAAAAAAAKi8/VK09R0E_FNkQ6QcoNJ5ERoYijtZvVnbHwCLcB/s1600/image011.gif)
Ainsi,
![](https://3.bp.blogspot.com/-fh9ctrdTo-U/VzIOTxFPyII/AAAAAAAAKjA/se4q66QkDRYSvz2DWmS1QKkZNNQ8DE_KACLcB/s1600/image012.gif)
5) a)
![](https://4.bp.blogspot.com/-KOT35iEXkdo/VzIOT7jpFSI/AAAAAAAAKjE/kpYx_h9uyHk7AQnyv42gMoKXPtUMM4CbQCLcB/s1600/image013.gif)
![](https://3.bp.blogspot.com/-uKzDfBD2IPo/VzIOUHtC01I/AAAAAAAAKjI/fCmhiWt_coMh4Xs04v53iA_Rf8m5jE6SACLcB/s1600/image014.gif)
![](https://4.bp.blogspot.com/-3AdNPxAG52Y/VzIOUt55cgI/AAAAAAAAKjQ/IgTAmjKy7D80mNlkKoL6bgCtwKN-4pH0QCLcB/s1600/image015.gif)
ainsi, E est continu en ![](https://3.bp.blogspot.com/-osr7Z6CWvvQ/VzIOUkPzGGI/AAAAAAAAKjM/68JLY5GAsas0ESoneok_cOHAPsnPagBYgCLcB/s1600/image016.gif)
![](https://3.bp.blogspot.com/-osr7Z6CWvvQ/VzIOUkPzGGI/AAAAAAAAKjM/68JLY5GAsas0ESoneok_cOHAPsnPagBYgCLcB/s1600/image016.gif)
![](https://3.bp.blogspot.com/-JzM_4Mhi8Do/VzIOUtmpsQI/AAAAAAAAKjU/T9cXADZbv60eYJ2jxmFR2vAvHRXV894-ACLcB/s1600/image017.gif)
ainsi, E est continu en r = R
8) R → R
a) En considérant un cylindre de hauteur h, la conservation de la charge s’écrit :
![](https://1.bp.blogspot.com/-Mt18JU32Uhk/VzIOVLsFbVI/AAAAAAAAKjY/N9jGtfU752gIqCz-B8tydVGdWs3czHUogCLcB/s1600/image018.gif)
D’où :
![](https://2.bp.blogspot.com/-AhJpVH2I6-s/VzIOVJQ3ejI/AAAAAAAAKjc/NjCrvRpib3Q3BLYMizQk1J8BQucO0L-TACLcB/s1600/image019.gif)
On retrouve le résultat de 2-b).
![](https://4.bp.blogspot.com/-WppovPLc7UQ/VzIOVWpWnYI/AAAAAAAAKjg/faK7mfYSgkg6HSva6tuRLnLUsSkwZYD0wCLcB/s1600/image020.gif)
![](https://2.bp.blogspot.com/-vGXQUN7k3X4/VzIOVl6g1YI/AAAAAAAAKjk/9z6FRzrwxq8q8TcbvMUoL6Vi2zDqernpQCLcB/s1600/image021.gif)
* Le fil passant par B de densité + λ crée le potentiel :
![](https://4.bp.blogspot.com/-VeR7XLrman0/VzIOVqin2QI/AAAAAAAAKjo/g1gM1y1sK507Ir1B2uYiumYBQEe0w81RACLcB/s1600/image022.gif)
* Le fil passant par A de densité - λ crée le potentiel :
![](https://2.bp.blogspot.com/-IadoPIu-AUs/VzIOVs7hm_I/AAAAAAAAKjs/OMnZZNer3YUJV0PjGQk4xBaAyMdrC2joQCLcB/s1600/image023.gif)
11) V (O) = 0
Le principe de superposition s’écrit :
![](https://3.bp.blogspot.com/-EUWKzWOcf5k/VzIOWfymq3I/AAAAAAAAKjw/7AD-3lL4euc5L8lhJWM738xbxwf4TNABACLcB/s1600/image024.gif)
![](https://3.bp.blogspot.com/-eVOAl3Uruk8/VzIOWqvP-8I/AAAAAAAAKj0/d6YK4fFn42419tnGsWeSQcz3ttbpgBNqQCLcB/s1600/image025.gif)
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