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Electric flow concept
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Flow is defined by the electric field and the section that the field lines traverse:
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Electric flow geometry
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The field is not necessarily orthogonal to the section for which the flow is being calculated. Therefore it is necessary to calculate that component of the electric field that is orthogonal to the section:
\\n\\nAs the dot product of a vector with a versor, in this case the versor that defines the orientation of the section, is the projection on it, it is obtained that the component to consider is the dot product or the cosine of the angle
$ \hat{n} \cdot \vec{E} =| \vec{E} | \cos \alpha$
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Flow by non-flat section and variable field
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As the section is not necessarily flat, the normal versor may vary its orientation. Similarly, the field can vary in direction and magnitude on the section. Therefore, the section can be segmented into small surface elements that can be considered flat at first approximation and in which the field does not vary either in direction or in magnitude:
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Electric field projection
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El producto punto de dos vectores se puede calcular con los módulos de los vectores y el angulo entre estos por lo que con
| $ \vec{a}\cdot\vec{b} = \mid\vec{a}\mid \mid\vec{b}\mid \cos \theta $ |
En el caso del versor su modulo es la unidad y se obtiene que con
| $ \vec{E} \cdot \hat{n} =\mid \vec{E} \mid\cos \alpha $ |
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Electric flux
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Si uno desea calcular el flujo por una sección que no necesariamente es plana y en que el campo no es constante se puede segmentar la sección en subsecciones
De esta forma, con el flujo sera la suma de las distintas contribuciones:
| $ \Phi = \displaystyle\sum_i \vec{E}_i\cdot\hat{n}_i\,dS_i $ |
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General electric flux
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La suma de flujos parciales corresponden con electric flow $N m^2/C$, surface electric field i $V/m$, surface element i $m^2$ and versor normal to surface i $-$
| $ \Phi = \displaystyle\sum_i \vec{E}_i\cdot\hat{n}_i\,dS_i $ |
se puede llevar al continuo pasando de la suma a una integral sobre la sección
| $ \Phi = \displaystyle\int \vec{E} \cdot \hat{n} dS $ |
con electric flow $N m^2/C$, surface electric field i $V/m$, surface element i $m^2$ and versor normal to surface i $-$.
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Electric flow, depending on the angle
Equation
If the flux is calculated on the different elements of the
| $ \Phi = \displaystyle\sum_i \vec{E}_i\cdot\hat{n}_i\,dS_i $ |
the expression with the angle between surface versor and
| $ \vec{E} \cdot \hat{n} =\mid \vec{E} \mid\cos \alpha $ |
field can be written as
| $ \Phi = \displaystyle\sum_i | \vec{E}_i | \cos\alpha_i dS $ |
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General electric flow, depending on the angle
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La suma de flujos parciales es con electric flow $N m^2/C$, surface electric field i $V/m$, surface element i $m^2$ and surface normal electric field angle i $rad$
| $ \Phi = \displaystyle\sum_i | \vec{E}_i | \cos\alpha_i dS $ |
se puede llevar al continuo pasando de la suma a una integral sobre la sección
| $ \Phi = \displaystyle\int | \vec{E} | \cos \alpha dS $ |
con electric flow $N m^2/C$, surface electric field i $V/m$, surface element i $m^2$ and surface normal electric field angle i $rad$.
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Video: Flux