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M3P2
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d343df9d
Commit
d343df9d
authored
Apr 27, 2024
by
Claude Meny
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Update cheatsheet.fr.md
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12.temporary_ins/10.electrostatics-vacuum/20.causes-stationary-electric-field/20.overview/cheatsheet.fr.md
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d343df9d
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@@ -745,9 +745,18 @@ figure à faire
*$`\boldsymbol{\mathbf{\displaystyle E_M=\int_{P\in\mathcal{C}} dE_{P\rightarrow M,z}}}`$*
*
L'ensemble des points $
`P`
$ constituant le disque, de coordonnées $
`P=(\rho_M,\,\varphi_M,\,0)`
$, s'obtient
en faisant varier
*$`\varphi_P`$ entre $`0`$ et $`2\pi`$*
et
*$`\rho_P`$ entre $`0`$ et $`R`$*
<br>
en faisant varier
*
*$`\varphi_P`$ entre $`0`$ et $`2\pi`$*
*
*$`\rho_P`$ entre $`0`$ et $`R`$*
Il s'agira donc de réaliser une
**intégrale double**
, dont les variables d'intégration
$
`d\varphi`
$ et $
`d\rho`
$ varient indépendamment l'une de l'autre.
*
Pour conduire la plus difficile des intégrations, celle relative à la variable $
`\rho`
$,
réécrivons le champ élémentaire en faisant disparaître le dénominateur pour l'exprimer au numérateur :
<br>
**$`\boldsymbol{\mathbf{dE_M}}`$**
$
\d
frac{
\d
ens^{2D}}{2
\e
psilon_0}
\c
dot
\d
frac{
\r
ho_P
\,
z_M}{(
\r
ho_P^2+z_M^2)^{
\,
3/2}}}}
`$
<br>
`
**$`\boldsymbol{\mathbf{\hspace{2.3cm}=\dfrac{\dens^{2D}\,z_M}{2\epsilon_0}\cdot\rho_P\,\(\rho_P^2+z_M^2)^{\,-3/2}}}}`$**
...
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