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Static Electric Field - Laplace s equation in 2D

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**1. **Static Electric Field - Laplace’s equation in 2D Solved by relaxiation In this project, I present a study of the two-dimensional static electric field problem for several cases.
The problem is formulated as Laplace's equation with boundary conditions.
The Laplace's equation is solved numerically by finite difference method and relaxation method.In this project, I present a study of the two-dimensional static electric field problem for several cases.
The problem is formulated as Laplace's equation with boundary conditions.
The Laplace's equation is solved numerically by finite difference method and relaxation method.

**2. **Static Electric Field Maxwell equations Reduced for static electric field problem Any classical electromagnetic problems are described by the set of Maxwell's equations
We neglect the magnetic field here.
For static electric field problem, the set of Maxwell's equations is reduced to the Poisson's equation.
The Poisson's equation is responsible for the space with charge distribution.
For free space, it further reduced to the Laplace's equation.Any classical electromagnetic problems are described by the set of Maxwell's equations
We neglect the magnetic field here.
For static electric field problem, the set of Maxwell's equations is reduced to the Poisson's equation.
The Poisson's equation is responsible for the space with charge distribution.
For free space, it further reduced to the Laplace's equation.

**3. **Problem 1 As shown in the picture The first problem is shown in the figure.
The z direction is not shown, which is perpendicular to the graph plane.
We have two grounded plane electrodes, one at y = 0 and another one at y = b.
They lie parallel to the xz plane and extend along the x direction from 0 to infinity.
The third electrode at x = 0 is maintained at potential V0.
The problem is described by Laplace's equation with the boundary conditions in two dimension.
We want to work out the potential in the area.
We shall set units b = 1 and V0 = 1, which means x and y are expressed in unit of b, and the potential in unit of V0.The first problem is shown in the figure.
The z direction is not shown, which is perpendicular to the graph plane.
We have two grounded plane electrodes, one at y = 0 and another one at y = b.
They lie parallel to the xz plane and extend along the x direction from 0 to infinity.
The third electrode at x = 0 is maintained at potential V0.
The problem is described by Laplace's equation with the boundary conditions in two dimension.
We want to work out the potential in the area.
We shall set units b = 1 and V0 = 1, which means x and y are expressed in unit of b, and the potential in unit of V0.

**4. **Analytical solution Fortunately, analytical solution exists for this problem.
The analytical solution is a solution in series form.
We will compare our numerical solution with this analytical one.
Let's see how many terms are enough to produce reasonable results.
We check it at the boundary x = 0, where the potential should equal 1.
The two graphs are obtained by sum of different number of terms, m_max.Fortunately, analytical solution exists for this problem.
The analytical solution is a solution in series form.
We will compare our numerical solution with this analytical one.
Let's see how many terms are enough to produce reasonable results.
We check it at the boundary x = 0, where the potential should equal 1.
The two graphs are obtained by sum of different number of terms, m_max.

**5. **Analytical solution We see 400 terms are enough.
Let's see the potential outline in the whole region.
The electrode at x = 0 maintain the largest value of potential.
The potential decrease with the x increasing.
Please remember this graph. We will compare it with our numerical solution later.We see 400 terms are enough.
Let's see the potential outline in the whole region.
The electrode at x = 0 maintain the largest value of potential.
The potential decrease with the x increasing.
Please remember this graph. We will compare it with our numerical solution later.

**6. **Numerical solution Now we solve the problem by numerical method.
First, we reformulate the problem by finite difference method.
We impose a grid of points on the semi-infinite rectangular region.
The griding is somewhat arbitrary. You may design your own griding scheme.
And you can choose a reasonable value of the grid spacing.
The boundary condition should also be reformulated.This is a finite difference replacement.
It leads to the expression, which means the potential value of the center can be expressed by the values of its neighbors.
To solve this problem, relaxation methods are anticipated.Now we solve the problem by numerical method.
First, we reformulate the problem by finite difference method.
We impose a grid of points on the semi-infinite rectangular region.
The griding is somewhat arbitrary. You may design your own griding scheme.
And you can choose a reasonable value of the grid spacing.
The boundary condition should also be reformulated.This is a finite difference replacement.
It leads to the expression, which means the potential value of the center can be expressed by the values of its neighbors.
To solve this problem, relaxation methods are anticipated.

**7. **Numerical solution Relaxation methods are iteration schemes designed.
Of all the relaxation methods, the Jacobi relaxation method and successive overrelaxation (SOR) method are often used.
By the Jacobi relaxation method, the iteration scheme is set up like this.
While for the SOR method, the iteration scheme is a little complicated.
You have freedom to choose the relaxation parameter w. It may accelerate the convergence of the iteration procedure.
We shall apply the SOR method to our problems.
The iteration starts from a guess of initial potential.
At the final step, the iteration will cease when a tolerance is reached.
We would have a max iteration, which can tell us how fast the relaxation procedure is.Relaxation methods are iteration schemes designed.
Of all the relaxation methods, the Jacobi relaxation method and successive overrelaxation (SOR) method are often used.
By the Jacobi relaxation method, the iteration scheme is set up like this.
While for the SOR method, the iteration scheme is a little complicated.
You have freedom to choose the relaxation parameter w. It may accelerate the convergence of the iteration procedure.
We shall apply the SOR method to our problems.
The iteration starts from a guess of initial potential.
At the final step, the iteration will cease when a tolerance is reached.
We would have a max iteration, which can tell us how fast the relaxation procedure is.

**8. **Now we set some reasonable values for the parameters.
(we have the relaxation parameter w ranging from 1.0 ~1.9, the grid spacing is about 1/20, the tolerance.)
They will affect the convergence speed.
This is a guess of the potential, which is inputted as an initial value.
Then we obtain the numerical result.
We see that the numerical outline is almost the same as the analytical one.
We have set different relaxation parameter w to observe the convergence speed of the relaxation.
We found the optimal value for the relaxation parameter is about 1.6 in our case.
This is the Mathematica code used in the computation.
Now we set some reasonable values for the parameters.
(we have the relaxation parameter w ranging from 1.0 ~1.9, the grid spacing is about 1/20, the tolerance.)
They will affect the convergence speed.
This is a guess of the potential, which is inputted as an initial value.
Then we obtain the numerical result.
We see that the numerical outline is almost the same as the analytical one.
We have set different relaxation parameter w to observe the convergence speed of the relaxation.
We found the optimal value for the relaxation parameter is about 1.6 in our case.
This is the Mathematica code used in the computation.

**9. **Problem 2 Let's see a similar problem. Now we have no analytical solution.
Likewise, we impose a grid of point in the rectangular region.
Notice that the boundary conditions are changed.
Let's set some reasonable values for the parameters.
The guess of the potential is the same as in the previous problem.
Then we work out the potential outline numerically.
We found the optimal value of the relaxation parameter is about 1.7.
Here the relaxation speed is very slow due to the bad guess of the potential.Let's see a similar problem. Now we have no analytical solution.
Likewise, we impose a grid of point in the rectangular region.
Notice that the boundary conditions are changed.
Let's set some reasonable values for the parameters.
The guess of the potential is the same as in the previous problem.
Then we work out the potential outline numerically.
We found the optimal value of the relaxation parameter is about 1.7.
Here the relaxation speed is very slow due to the bad guess of the potential.

**10. **Problem 3 The third problem is about a triangular region as shown in the figure.
We have only two electrodes. One is grounded, another is maintained at potential V0.
First, we impose a grid of points in the triangular region.
Then set up the boundary conditions.
And then give a guess of potential.
So the problem is solved to give the potential outline shown in the figure.
We also observed the convergence speed of the relaxation.
The optimal value for the relaxation parameter is among 1.2~1.8.The third problem is about a triangular region as shown in the figure.
We have only two electrodes. One is grounded, another is maintained at potential V0.
First, we impose a grid of points in the triangular region.
Then set up the boundary conditions.
And then give a guess of potential.
So the problem is solved to give the potential outline shown in the figure.
We also observed the convergence speed of the relaxation.
The optimal value for the relaxation parameter is among 1.2~1.8.

**11. **Prospect We have successively solved the Laplace's equation in two dimension by numerical method.
The numerical method consists of the finite difference method and the relaxation method.
We would expect that the numerical method is very useful in solving the problems with irregular boundaries as shown in the figure.We have successively solved the Laplace's equation in two dimension by numerical method.
The numerical method consists of the finite difference method and the relaxation method.
We would expect that the numerical method is very useful in solving the problems with irregular boundaries as shown in the figure.

**12. **End Thanks