Gauss' law in line integral, Q=##ϵ_0 ∮E.n dl=-ϵ_0 ∮∂ϕ/∂n dl##

[mdn]

I know the Gauss law for surface integral to calculate total charge by integrating the normal components of electric field around whole surface . but in above expression charge is calculated using line integration of normal components of electric field along line. i don't understand this relation. any help please.

 

[pasmith]

Is this a 2D problem? Because integrating over a closed curve (ie the "surface" of an area) is the 2D equivalent of integrating over the surface of a volume. By Green's Theorem, $$\begin{split} \int_{\Omega} \nabla \cdot \mathbf{E}\,dA &= \oint_{\partial\Omega} (-E_y, E_x, 0)\cdot \mathbf{t}\,dl \\ &= \oint_{\partial \Omega} (\mathbf{k} \times \mathbf{E}) \cdot \mathbf{t}\,dl \\ &= \oint_{\partial \Omega} \mathbf{E} \cdot (\mathbf{t} \times \mathbf{k})\,dl \\ &= \oint_{\partial \Omega} \mathbf{E} \cdot \mathbf{n}\,dl \end{split}$$ since $\mathbf{k} = \mathbf{n} \times \mathbf{t}$ where $\mathbf{t}$ is the unit tangent of the curve traversed anticlockwise and $\mathbf{n}$ is the outward unit normal.

 

[mdn]

Is this a 2D problem? Because integrating over a closed curve (ie the "surface" of an area) is the 2D equivalent of integrating over the surface of a volume. By Green's Theorem, $$\begin{split} \int_{\Omega} \nabla \cdot \mathbf{E}\,dA &= \oint_{\partial\Omega} (-E_y, E_x, 0)\cdot \mathbf{t}\,dl \\ &= \oint_{\partial \Omega} (\mathbf{k} \times \mathbf{E}) \cdot \mathbf{t}\,dl \\ &= \oint_{\partial \Omega} \mathbf{E} \cdot (\mathbf{t} \times \mathbf{k})\,dl \\ &= \oint_{\partial \Omega} \mathbf{E} \cdot \mathbf{n}\,dl \end{split}$$ since $\mathbf{k} = \mathbf{n} \times \mathbf{t}$ where $\mathbf{t}$ is the unit tangent of the curve traversed anticlockwise and $\mathbf{n}$ is the outward unit norm

yes, this is the 2D problem. i am trying to calculate the total charge on conductor, shown by bold lines in 2D

domain. i have calculated potential distribution at each point using Laplace equation in finite element method, now i want to calculate total charge Q on this conductor, from this charge i want to calculate the capacitance. i want to use gauss divergence theorem to calculate charge $E\bar =-\nabla\phi$, where $\phi =potential$. here is my real problem. how to calculate total charge on conductor in above case and which normal components of electric field (in terms of scalar potential $\nabla\phi$) should i integrate to get total charge?

from my side i have written equation like this..
$Q=\oint (E. n) dl$
Q=$\nabla\phi .n$
=$\oint(\partial\phi/\partial x +\partial \phi/\partial y)$ dl
=$\oint {\partial \phi/ \partial x} dl +\oint {\partial \phi/ \partial y} dl$
=$\oint {\partial \phi/ \partial x} dy +\oint {\partial \phi/ \partial y} dx$
is this correct?