The twelfth time homework——Potentials and Fields

谭善

2014301020106

---

---

1. Abstract

---

---

In physics, it is inevitable to make contact with Potentials and Fields. In this part, I will show you some interesting situations of them.

Question 5.1

Sovle for the potential in the prism geometry in Figure 5.4.

Question 5.4

Investigate how the magnetitude of the fringing field of a parallel plate capacitor, that is, the electric field outside the central region of the capacitor in Figure 5.6, varies as a function of the plate separation.

Question 5.6

Calculate the electric potential and field near a lightning rod. Model this as a very long and narrow metal rod held at a high voltage, with one end near a conducting plane . Of special interest is the field near the tip of the rod.

---

---

2. Background and Introduction

---

---

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

At the point (i,j,k) the derivative with respect to x may be written as

Then, it is natural to write the second partial derivative as

This expression is nicely symmetric in the way it treats V(i+1,j,k) and V(i-1,j,k), which will trun out to reduce the overall errors in our computations. The results for the other second partial derivatives have similar forms. Inserting them all into Laplace's equation and solving for V(i,j,k) we find

As the same as below, when we deal with a two-dimensional problem, our goal is to find the potential fuction V(i,j) that satisfies

In order to find out the electric field, we have the equation

---

---

3. Body Content

---

---

• Question 5.1-the prism geometry

We plot the prism geometry diagram of the potential as below. Of course, I also plot its electri field.

prism geometry diagram

• Question 5.3-the magnitude of the fringing field of a parallel plate capacitor.

First, we plot its potential in different ways.

Latter, let's consider its electric field.

The following figures are given by the plate separation are 0.6m, 0.8m, 1.0m, 1.2m.

We will find that when the plate separation is larger, the magnitude of the fringing field of a parallel plate capacitor is larger.

• Question 5.6

For this question, we assume when the electric potential and field is zero at infinity. Then, we can obtain

From figure in the middle, we find that it is just like a baseball bat, that, the electric field of the end that is near conducting plane is very large when the other end is small.

• Electric charges

For one point, we can plot the following figure.

It is just like one atom's potential.

If we move some atom together, we find that

It is quite helpful for us to study lattice, we can use it into solid state physics.

• Saddle surface

We just cann't stop plot these picture, they are amazing, especially for saddle surface.

---

---

4. Conclusion

---

---

• In the process of soving problem, the most important thing is to consider its boundary condition.

• We must ensure that \Delta V is very small, otherwise, what we ploted is almost totally different from the real case.

• After ploting these figures, I think this method can be applied to solid state physics.

---

---

5. Reference and Acknowledgement

• Computational Physics (Second Edition), Nicholas J. Giordano, Hisao Nakannishi.

• How to plot colorful figure.

• Thanks to Wuyuqiao

---

---

6. Code

• Code-potential

• Code-field