I have evaluated the 3D potential distributions between parallel plates of a disc capacitor .I used boundary condition as 1V-0V for solving laplace equation. I calculated the capacitance from geometrical equation. I want to compare the values of capacitance from my model using various meshing strategies. I have V as column vector for all points within the plates and at the plates. how can compute the charge induced at one plate or total energy stored to compute the capacitance?
For 3D model capacitance calculation used ansys Maxwell or comsol multiphysics software.
I have done exactly the same as you and have the same qzestion: Given the potential distribution on a grid of points, what is the next step in the procedure to calculate the capacitance?
I guess we have to integrate something like D around a closed contour to find the charge. But D is a vector and charge is a scalar. And D.ds is going to be zero as D is almost normal to the path of integration
Hope you got answer to your question. I have now figured out what to do.
You put a box around one of the conductor, e.g. the smallest, and calculate the electric field on its boundary by using for examply this finite difference formula to get the grad from the potential:
X-component of grad at grid indices x,y,z is [8(V(x+1) - V(x-1) -(V(x+2)-V(x-2)) ]/12
Y and Z components may be likewise obtained by using 4 points along those dimensions around (x,y,z)
Now you add up only those electric field components NORMAL to the box side and select the sign to be negative if the normal is pointing in the opposite direction to the field, because you want the sum of the flux D coming OUT of the box.
For accuracy, the corners might need to be treated differently. In 2D the corner points contribute both an Ex and an Ey component so
I figured out they needed to be weighted by 0.5 In 3D points on the sides of the box need to be weighted by 0.5 and corners by 0.25
Once you have the total flux of E coming out of the box normal to its sides, multiply by epsilon and you have the capacitance.
Paul Dent Thank you for your answer. I computed the capacitance from electric flux density and charges induced on positive plate
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