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# System prepended metadata

title: Chapter 5 - Electric Field in Material Space

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[TOC]
# Chapter 5 - Electric Field in Material Space
## current in conductors
:::spoiler current in conductors
![](https://i.imgur.com/qgPDCb0.png)
![](https://i.imgur.com/RauH9D3.png)
:::

## current 有兩種
:::spoiler 1. conduction current
> ![](https://i.imgur.com/whpR87X.png)
> ![](https://i.imgur.com/JUo4Klx.png)
> ![](https://i.imgur.com/oOzNYVB.png)
:::

:::spoiler 2. convection current 對流電流
> ![](https://i.imgur.com/OGrAIWG.png)
:::

## Conductor
![](https://i.imgur.com/5h6lGBp.png)

### Resistivity of Conductors
![](https://i.imgur.com/Gpw8S4u.png)
![](https://i.imgur.com/GswmGIi.png)

### Joule's Law
:::spoiler Power
![](https://i.imgur.com/wLybszC.png)
:::
:::spoiler Power Density
![](https://i.imgur.com/qczcRxL.png)
:::
![](https://i.imgur.com/QcvT3BL.png)

## Continuity Equation
![](https://i.imgur.com/ajslSal.png)

## Relaxation Time
![](https://i.imgur.com/W6oip5r.png)

# Chapter 6 - Electrostatic Boundary-Value Problems

Poisson's and Laplace's Equations
得到電場的三種方法
:::spoiler rho (向量場)
![](https://i.imgur.com/bTJSUzG.png)
:::
:::spoiler 高斯 (向量場)
![](https://i.imgur.com/egJtxxg.png)
:::
:::spoiler V (純量場)
![](https://i.imgur.com/X0wRdcx.png)
:::

但是有兩種限制
:::spoiler 
![](https://i.imgur.com/4GJiJln.png)
:::
所以我們就要進入 Boundary-Value Problems
首先，從 Poisson's equation and Laplacian equation 開始
## Poisson's equation
:::spoiler Poisson's equation
![](https://i.imgur.com/UM6I2tx.png)
:::
## Laplacian equation
:::spoiler Laplacian equation (Poisson's equation 的特例)
![](https://i.imgur.com/IC6n1Dz.png)
:::
## Uniqueness theorem
:::spoiler Uniqueness theorem
![](https://i.imgur.com/YCwC7hz.png)
![](https://i.imgur.com/dBKC3y7.png)
![](https://i.imgur.com/RU16lBB.png)
![](https://i.imgur.com/pqzxIV2.png)
邊界條件常見型態
![](https://i.imgur.com/FzmY8iJ.png)
:::

## Solving Poisson's or Laplace's equations
:::spoiler
![](https://i.imgur.com/xcut1xc.png)
:::

## Method of separable variables
![](https://i.imgur.com/BSGazYC.png)
然後討論 lamda>=<0 的情況
:::spoiler lamda>0
![](https://i.imgur.com/Ma2hNNo.png)
:::
:::spoiler lamda=0
![](https://i.imgur.com/EO98DuD.png)
:::

## Solutions for 2D Laplace's equations
Cartesian Coordinate System
![](https://i.imgur.com/0C16UmQ.png) beta<0，和beta>0，式子長的一樣
![](https://i.imgur.com/EUAJ5iO.png)
所以 V 有 2X2 種型態
或是
![](https://i.imgur.com/72bsNqZ.png)
所以 V 有 2X2 種型態
所以
![](https://i.imgur.com/dHBNEuZ.png)
![](https://i.imgur.com/IMmITjt.png)
注意：要注意 possible complete solutions 不要寫成 invalid solutions

# 心得
這段雖然數學偏重看似困難，但是實質上如果我們把過程看過，然後知道了過程以後忽視它，只記得她的結果的話，那麼就沒有想像中那麼難，主要是寫題目比較難。很開心最近稍微有一點漸入佳境的感覺了，加油 fighting！



























































































































