I need to solve the Laplace Differential Equation Del²Φ = 0 and find the electric potential at different points due to a charged plate with uniform surface charge density λ/ m².
Dear colleagues, can you please help me with your valuable suggestions?
Thanks and Regards
N Das
Dear Nityananda, I would suggest a simple finite difference scheme using Taylor expansion.
-> Laplace-Equation (2D):
0=(∂^2 Φ )/(∂x^2 )+(∂^2 Φ )/(∂y^2 )
with:
(∂^2 Φ)/(∂x^2 )=(Φ_(i+1,j)-2Φ_(i,j)+Φ_(i-1,j))/(Δx^2)
(∂^2 Φ)/(∂y^2 )=(Φ_(i,j+1)-2Φ_(i,j)+Φ_(i,j-1))/(Δx^2)
which leads to:
0 = (Φi+1,j − 2Φi,j + Φi−1,j)/(∆x^2) +
(Φi,j+1 − 2Φi,j + Φi,j−1)/(∆y^2)
with the same spatial discretization ∆x=∆y this is simplified to
0 = 4Φi,j − Φi−1,j − Φi,j−1 − Φi+1,j − Φi,j+1
Dear Nityananda Das,
If you are Wolfram Mathematica user, you may be interested in NDSolve or NDSolveValue functions:
https://mathematica.stackexchange.com/questions/191984/not-sure-how-to-set-up-the-laplacian-poisson-equation
https://reference.wolfram.com/language/ref/NDSolve.html
https://reference.wolfram.com/language/ref/NDSolveValue.html
Respected Marco Brysch,
Thanks for your suggestion.
I am trying.
Regards
N Das
Respected Victor J Adlucky,
Thanks a lot.
But, I don't use Mathematica.
I use Matlab.
Is it possible by Matlab ?
Regards
N Das
Might want to look at a classic like Courant and Hilbert, Methods of Mathematical Physics. Most likely this problem can be solved by separation of variables - reducing the problem to a set of ODE which can be then solved.
Dear Nityananda Das,
For regular spatial domains you can try the separation of variables method for problems that may provide an analytical solution or, where only a numerical solution is possible, the finite-element/finite-difference/finite-volume methods are available. You will find a large body of literature available in books, papers and on the Internet detailing all these methods.
The situation becomes more difficult when the problem involves irregular spatial domains. For these types of problems a numerical approach such as the finite-element method or meshless method are a good way forward. If you are using Matlab, then using the 'finite-element toolbox' is straight forward. However, if you are programming up the solution yourself, the meshless method is far easier to implement and usually provides good results.
I hope the above is useful.
Kind regards,
Graham W Griffiths
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