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M3P2
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92069ab7
Commit
92069ab7
authored
Nov 14, 2019
by
Claude Meny
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92069ab7
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...
@@ -616,47 +616,47 @@ Intégral (magnétostatique + électromagnétisme)
$
`\displaystyle\oiint_S\vec{B}\cdot\vec{dS}=0`
$
$
`\displaystyle\oint_{\Gamma\,orient.}\
vec{B} \cdot \vec
{dl}=
$
`\displaystyle\oint_{\Gamma\,orient.}\
overrightarrow{B} \cdot \overrightarrow
{dl}=
\mu_0\underset{S\,orient.}{\sum{\overline{\,I\,}}}`
$
$
`\displaystyle\oint_{\Gamma\,orient.}\
vec{B} \cdot \vec
{dl}=
\mu_0\underset{S\,orient.}{\iint{\
vec{j}\cdot\vec
{dS}}}`
$
$
`\displaystyle\oint_{\Gamma\,orient.}\
overrightarrow{B} \cdot \overrightarrow
{dl}=
\mu_0\underset{S\,orient.}{\iint{\
overrightarrow{j}\cdot\overrightarrow
{dS}}}`
$
$
`\displaystyle\oint_{\Gamma\,orient.}\
vec{H} \cdot \vec
{dl}=
$
`\displaystyle\oint_{\Gamma\,orient.}\
overrightarrow{H} \cdot \overrightarrow
{dl}=
\underset{S\,orient.}{\sum{\overline{\,I\,}}}`
$
$
`\displaystyle\oint_{\Gamma\,orient.}\
vec{H} \cdot \vec
{dl}=
\underset{S\,orient.}{\iint{\
vec{j}\cdot\vec
{dS}}}`
$
$
`\displaystyle\oint_{\Gamma\,orient.}\
overrightarrow{H} \cdot \overrightarrow
{dl}=
\underset{S\,orient.}{\iint{\
overrightarrow{j}\cdot\overrightarrow
{dS}}}`
$
local (magnétostatique)
$
`
rot\vec{B}=\mu_0 \cdot \vec
{j}`
$
$
`
\overrightarrow{rot}\overrightarrow{B}=\mu_0 \cdot \overrightarrow
{j}`
$
Electromagnétisme dans le vide :
$
`
rot\vec{B}=\mu_0 \cdot \vec{j}\,+ \, \epsilon_0\mu_0 \cdot \dfrac{\partial \vec{E}}{\partial t}`
$$
`=\mu_0 \cdot \vec{j}\,+ \, \dfrac{1}{c^2} \cdot \dfrac{\partial \vec{E}}{\partial t}`
$$
`=\mu_0 \cdot \vec{j}\,+ \mu_0 \cdot \vec
{j_D} = \mu_0 \cdot (j+j_D)`
$
$
`
\overrightarrow{rot}\overrightarrow{B}=\mu_0 \cdot \overrightarrow{j}\,+ \, \epsilon_0\mu_0 \cdot \dfrac{\partial \overrightarrow{E}}{\partial t}`
$$
`=\mu_0 \cdot \overrightarrow{j}\,+ \, \dfrac{1}{c^2} \cdot \dfrac{\partial \overrightarrow{E}{\partial t}`
$$
`=\mu_0 \cdot \overrightarrow{j}\,+ \mu_0 \cdot \overrightarrow
{j_D} = \mu_0 \cdot (j+j_D)`
$
avec $
`\
vec{j_D}`
$ courant de déplacement : $
`\vec{j_D}=\epsilon_0 \cdot \dfrac{\partial \vec
{E}}{\partial t}`
$
avec $
`\
overrightarrow{j_D}`
$ courant de déplacement : $
`\overrightarrow{j_D}=\epsilon_0 \cdot \dfrac{\partial \overrightarrow
{E}}{\partial t}`
$
Con corriente de desplazamiento
$
`\
vec{rot}\vec{E}=-\dfrac{\partial \vec
{B}}{\partial t}`
$
$
`\
overrightarrow{rot}\overrightarrow{E}=-\dfrac{\partial \overrightarrow
{B}}{\partial t}`
$
$
`div\
vec
{E}=\dfrac{\rho}{\epsilon_0}`
$
$
`div\
overrightarrow
{E}=\dfrac{\rho}{\epsilon_0}`
$
$
`\
vec{D}=\epsilon \vec{E} = \epsilon_0 \epsilon_r \vec
{E} `
$
$
`\
overrightarrow{D}=\epsilon \overrightarrow{E} = \epsilon_0 \epsilon_r \overrightarrow
{E} `
$
Propriétés anisotropes :
$
`\
vec
{D}= \overrightarrow{\overrightarrow{
\epsilon}}\, \
vec
{E}= \epsilon_0 \, \overrightarrow{\overrightarrow{
\epsilon_r}} \, \
vec
{E}`
$
$
`\
overrightarrow
{D}= \overrightarrow{\overrightarrow{
\epsilon}}\, \
overrightarrow
{E}= \epsilon_0 \, \overrightarrow{\overrightarrow{
\epsilon_r}} \, \
overrightarrow
{E}`
$
-
si P est dans le vide : $
`\
vec{D}=\epsilon_0 \cdot \vec
{E}`
$
-
si P est dans le vide : $
`\
overrightarrow{D}=\epsilon_0 \cdot \overrightarrow
{E}`
$
-
si P est dans un milieu diélectrique (homogène et isotrope)
$
`\
vec{D}=\epsilon \cdot \vec{E} = \epsilon_0 \cdot \epsilon_r \cdot \vec
{E} `
$
$
`\
overrightarrow{D}=\epsilon \cdot \overrightarrow{E} = \epsilon_0 \cdot \epsilon_r \cdot \overrightarrow
{E} `
$
avec $
`\epsilon`
$ : permittivité électrique absolue du milieu
$
`\epsilon_r`
$ : permittivité électrique absolue du milieu
...
...
@@ -676,24 +676,24 @@ $`\epsilon_0 \cdot \mu_0 \cdot c^2 = 1`$
#### Ecuaciones de Maxwell en forma diferencial / Equations de maxwell locales / ...
$
`div\
vec
{E}=\dfrac{\rho}{\epsilon_0}`
$
$
`div\
overrightarrow
{E}=\dfrac{\rho}{\epsilon_0}`
$
$
`rot\
vec{E}=-\dfrac{\partial \vec
{B}}{\partial t}`
$
$
`rot\
overrightarrow{E}=-\dfrac{\partial \overrightarrow
{B}}{\partial t}`
$
$
`div\
vec
{B}=0`
$
$
`div\
overrightarrow
{B}=0`
$
$
`rot\
vec{B}=\mu_0 \cdot \vec{j}\,+ \, \epsilon_0\mu_0 \cdot \dfrac{\partial \vec{E}}{\partial t}`
$$
`=\mu_0 \cdot \vec{j}\,+ \, \dfrac{1}{c^2} \cdot \dfrac{\partial \vec
{E}}{\partial t}`
$
$
`rot\
overrightarrow{B}=\mu_0 \cdot \overrightarrow{j}\,+ \, \epsilon_0\mu_0 \cdot \dfrac{\partial \overrightarrow{E}}{\partial t}`
$$
`=\mu_0 \cdot \overrightarrow{j}\,+ \, \dfrac{1}{c^2} \cdot \dfrac{\partial \overrightarrow
{E}}{\partial t}`
$
#### Ecuaciones de Maxwell en forma integral / Equations de maxwell intégrales / ...
$
`\displaystyle\oiint_S\
vec{E}\cdot\vec
{dS}=\dfrac{Q_{int}}{\epsilon_0}`
$$
`=\dfrac{1}{\epsilon_0} \cdot \iiint_{\tau \leftrightarrow S} \rho \cdot d\tau`
$
$
`\displaystyle\oiint_S\
overrightarrow{E}\cdot\overrightarrow
{dS}=\dfrac{Q_{int}}{\epsilon_0}`
$$
`=\dfrac{1}{\epsilon_0} \cdot \iiint_{\tau \leftrightarrow S} \rho \cdot d\tau`
$
$
`\displaystyle\oiint_S\
vec{B}\cdot\vec
{dS}=0`
$
$
`\displaystyle\oiint_S\
overrightarrow{B}\cdot\overrightarrow
{dS}=0`
$
$
`\displaystyle\iiint_{\tau} div\
vec
{E} \cdot d\tau= \displaystyle\iiint_{\tau} \dfrac{\rho}{\epsilon_0} \cdot d\tau = \dfrac{1}{\epsilon_0} \cdot \iiint_{\tau} \rho \cdot d\tau = \dfrac{Q_{int}}{\epsilon_0} `
$
$
`\displaystyle\iiint_{\tau} div\
overrightarrow
{E} \cdot d\tau= \displaystyle\iiint_{\tau} \dfrac{\rho}{\epsilon_0} \cdot d\tau = \dfrac{1}{\epsilon_0} \cdot \iiint_{\tau} \rho \cdot d\tau = \dfrac{Q_{int}}{\epsilon_0} `
$
$
`\displaystyle\iint_S \overrightarrow{rot} \,\overrightarrow{E}\cdot \overrightarrow{dS} = -\displaystyle\iint_{S \leftrightarrow \tau} \overrightarrow{B}\cdot \overrightarrow{dS}`
$
...
...
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