homework13(the field model)

作业

name:姚逸飞
student number：2013301020096

Homework

• $\Bbb homework\; L:$The field model

Abstract

The electrostatic potential is a very important question in pysics theory. This passage will discuss about it.
The electrostatic potential is obeyed to the partial differential equation ,for simplicity, as PED. There lots of method to solve the PED, like the separation of variable method. But in the caculation of physics, we still choose the Euler-Cromer method.

(Here is a magnetic field picture)

Passage

The issue of electrostatic potential without sources can be solved by the Laplace equation:

$$\Delta V=0$$
This equation can also be written as:
$$\frac{\partial^2 V}{\partial x^2}+\frac{\partial^2 V}{\partial y^2}+\frac{\partial^2 V}{\partial z^2}=0$$
In the text book ,we can get the separate form:
$$V(i,j,k)=\frac{1}{6}[V(i-1,j,k)+V(i+1,j,k)+V(i,j-1,k)+\\V(i,j+1,k)+V(i,j,k-1)+V(i,j,k+1)]$$
So from the separate equation,we can see the next value is determined by the six value around it. So we should figure out one value of the euqtion .Then we can put the value into the separate eequation. Then we can get all the value that we want.
We choose the plane parallel capacitor with $V=1$and$V=-1.$
Then we can plot the picture as follow:

This picture is about the plane parallel capacitor with $V=1$and$V=0.$

Here we discuss about the different steps in the two model :

Conclusion

We can see purely the $V$ from the picture.
If time permits, I'll discuss about the impact of different method.
The total codes are here

Reference

• matplotlib
• our text book
• latex