I am doing an investigation into how the capacitance (at the moment, of a parallel plate capacitor), differs when using idealistic vs realistic models i.e. the well known analytic solution: $C=\epsilon_0 \frac{A}{d}$ vs numerical methods.My set up is as follows: I have written a fortran 90 program that solved a finite difference approximation of the Laplace equation giving the potential distribution within a grounded box (outer boundaries set to constant potential 0) and parallel plates set to constant potential -1 and +1.'I' have derived that the ratio C_realistic/C_idealistic can be numerically calculated with the following algorithm:$$\frac{C}{C_{\infty}}=\frac{d}{V l} \sum_{bound} |\phi_{outer}-\phi_{plate}|$$ Where d is the distance between plates, V is the voltage (potential difference i.e. 2 here), l is the length of one of the plates and $\phi_{plate}$ is the potential at the edge of the plate and $\phi_{outer}$ is the potential of the point immediately 'outside' $\phi_{plate}$ I am investigating how this changes as i move the plates further away from each other, and what i have found is that, based on the results I'm getting, as the plates are moved further away from each other the ratio of realistic/idealistic increases, implying that the realistic electric field can store more energy. This does not seem correct to me so i have been testing my code and all values in the formula seem to have the right values and be behaving as they should. As the plates move away from each other, the difference in the potential between the points on the plates and those surrounding it decreases as expected, but d is increasing at a faster rate meaning the ratio is increasing.Anyone have any idea where i am going wrong, if anywhere?Thanks.(This is part of a degree project, so if this is in anyway cheating or whatever please don't answer :) )
Realistic/ Idealistic Capacitance ratio not as i expect
computational physics