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

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The discussion revolves around applying Gauss' law in a 2D context to calculate total charge on a conductor using line integrals of the electric field. The original poster seeks clarification on how to relate the line integral expression to charge calculation, particularly in the context of a 2D problem. They have computed the potential distribution using the finite element method and now aim to derive the total charge from the electric field. The conversation highlights the use of Green's Theorem to connect the divergence of the electric field to line integrals along the boundary. The poster proposes an equation for charge calculation, seeking validation on its correctness.
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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.
 
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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, <br /> \begin{split}<br /> \int_{\Omega} \nabla \cdot \mathbf{E}\,dA &amp;= \oint_{\partial\Omega} (-E_y, E_x, 0)\cdot \mathbf{t}\,dl \\<br /> &amp;= \oint_{\partial \Omega} (\mathbf{k} \times \mathbf{E}) \cdot \mathbf{t}\,dl \\<br /> &amp;= \oint_{\partial \Omega} \mathbf{E} \cdot (\mathbf{t} \times \mathbf{k})\,dl \\<br /> &amp;= \oint_{\partial \Omega} \mathbf{E} \cdot \mathbf{n}\,dl<br /> \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.
 
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pasmith said:
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, <br /> \begin{split}<br /> \int_{\Omega} \nabla \cdot \mathbf{E}\,dA &amp;= \oint_{\partial\Omega} (-E_y, E_x, 0)\cdot \mathbf{t}\,dl \\<br /> &amp;= \oint_{\partial \Omega} (\mathbf{k} \times \mathbf{E}) \cdot \mathbf{t}\,dl \\<br /> &amp;= \oint_{\partial \Omega} \mathbf{E} \cdot (\mathbf{t} \times \mathbf{k})\,dl \\<br /> &amp;= \oint_{\partial \Omega} \mathbf{E} \cdot \mathbf{n}\,dl<br /> \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
1664674830832.png

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?
 

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