HOMEWORK13——Electric Potential

homework

author : 2013301020086 Yuqi Wang

1.Abstract

This homework explored the electric potential with the bound condition in two-dimension.

2.Model

In regions of space that do not contain any electric charges, the electric potential obeys Laplaces's equation:

$$\frac{\partial^2V}{\partial x^2}+\frac{\partial^2V}{\partial y^2}+\frac{\partial^2V}{\partial z^2}=0$$
In two-dimensional case the equation reduces to:
$$\frac{\partial^2V}{\partial x^2}+\frac{\partial^2V}{\partial y^2}=0$$
$V$ is known at the boundaries.

3.Algorithm

We discretize the independent variables $x$ and $y$. Points in our space are then specified by integer $i$ and $j$, with $x=i\Delta x, y=j\Delta y$.
We find the potential function:

$$V(i,j)=\frac{1}{4}[V(i+1,j)+V(i-1,j)+V(i,j+1)+V(i,j-1)]$$
The approach we take to get the solution is to begin with some initial guess for the solution, $V_0(i,j)$, say. Then we use the formula above to obtain the new $V$. This procedure will be repeated until the difference between the new $V$ and old $V$ is sufficiently small.

4.Boundary Condition and Solution

This is the complete code

The picture following is the boundary condition:

This is the solution:

This is the code plotting the picture

5.Acknowledgement

2013301020084 Han Xu