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# System prepended metadata

title: Weekly Report 2.3

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# Weekly Report 2.3
[toc]

# Electrostatic Boundary-Value Problems
## Cartesian Coordinate System
![](https://i.imgur.com/qPQUWa3.png)
同樣的透過分離變數法
![](https://i.imgur.com/XQUGqBI.png)
然後寫出 Laplace Equation
![](https://i.imgur.com/QKqm02L.png)
然後假設他可以做分離
![](https://i.imgur.com/1orliv0.png)
接下來就分成四個 CASE：
:::spoiler k 都等於 0 (3linear)
![](https://i.imgur.com/WSdFlSp.png)
:::
:::spoiler 一個 k 等於 0 (1linear、1sincos、1sinhcosh)
![](https://i.imgur.com/bUu6AM2.png)
![](https://i.imgur.com/YwmnVrX.png)
:::
:::spoiler 兩實一虛 (2sincos、1sinhcosh)
![](https://i.imgur.com/xuPET29.png)
這個情況下才會滿足
![](https://i.imgur.com/8PSc9GW.png)
![](https://i.imgur.com/qPovb9M.png)
其中裡面的 k_z 是這樣來的 (只要記得是平方和開根號就行)
![](https://i.imgur.com/dt7L7zC.png)
:::
:::spoiler 一實兩虛 (1sincos、2sinhcosh)
長相可以參考 兩實一虛 (2sincos、1sinhcosh)，但是記得微調
![](https://i.imgur.com/qPovb9M.png)
:::
:::danger
透過對照整理來記憶
:::
## Cylindrical Coordinate
:::spoiler 解題順序
![](https://i.imgur.com/dZoTeyG.png)
然後就可以得到
![](https://i.imgur.com/8jUtqhx.png)
:::
## <補充> Bessel Function
:::spoiler J 的特性
![](https://i.imgur.com/8jtwTSz.png)
:::
:::spoiler N & Y 的特性
![](https://i.imgur.com/ro8gfCw.png)
:::

# Laplace’s Equations in Cylindrical Coordinates
## Potential Functions without z-dependence
### An infinitely long cylinder: a system with potential independent of z
這是一個特殊情況
因為
![](https://i.imgur.com/uHfLpwN.png)
所以
![](https://i.imgur.com/cfRQT5e.png)
原先看起來是 Bessel Function 的，變成不是了 
![](https://i.imgur.com/hyLWfTo.png)
所以
![](https://i.imgur.com/UNDW0PL.png)
:::danger
記後面的這兩個
:::

### A Coaxial Capacitor
![](https://i.imgur.com/EiGxVKX.png)
我們可以透過 Laplace 再來做他一次
![](https://i.imgur.com/DwksmxO.png) ![](https://i.imgur.com/PLVOUcr.png)
:::spoiler 自己盲寫一次
![](https://i.imgur.com/PrUp9gP.png)
:::

### A cylindrical surface at a with potential distribution in free space
![](https://i.imgur.com/X4JeFXv.png)
:::info
透過把cos換掉的方式，可以幫助我們在後面的時候可以用"比較係數"來得到我們要的東西
![](https://i.imgur.com/kctiuMs.png)
:::
![](https://i.imgur.com/73504fE.png)
:::spoiler 過程
![](https://i.imgur.com/pa1kItq.png)    ![](https://i.imgur.com/kkYmCFE.png)
:::
:::danger
要注意的是 C 和 D = 0 的原因
:::
![](https://i.imgur.com/56r0xEO.png)

## Potential Functions with z-dependence
:::danger
要小心他和前面 with independence 的不同之處 (晚一點讀完再整理出來)
:::
:::spoiler 同樣透過變數分離法，寫出三個式子
![](https://i.imgur.com/W7dhwLG.png)
![](https://i.imgur.com/aEtJorC.png)
![](https://i.imgur.com/paGefCb.png)
:::
![](https://i.imgur.com/nrVjiAv.png)
![](https://i.imgur.com/y5nBR1D.png)
:::spoiler I 的特性
![](https://i.imgur.com/ym8yrRs.png)
:::
:::spoiler K 的特性
![](https://i.imgur.com/bNXqaS4.png)
:::
:::spoiler 過程和之前類似，故省略
![](https://i.imgur.com/HPkl4o2.png)
![](https://i.imgur.com/yngczSW.png)
:::
![](https://i.imgur.com/BzM9LHB.png)

# Electrostatic Boundary-Value Problems
:::danger
此講義之內容考試高機率出現，課本裡面未必有，請務必詳讀
:::
## Method of Images 
![](https://i.imgur.com/QQ4o0zo.png)
:::spoiler The method of images: A Charge over a Grounded Plane
![](https://i.imgur.com/7RaEylP.png)
透過庫侖定律可以得到
![](https://i.imgur.com/4oZfBHv.png)
![](https://i.imgur.com/g5YRYJq.png)
:::
A Charge over a Grounded Conducting Plane 的詳細過程看影片

:::warning
當我們擺放 image 的時候要注意是要擺在 問題空間的對面
:::

前提是
1. 我們寫出來的 solution V 必須要滿足 Laplace 或是 Possion's Equation 
2. 檢查我們寫出來的 Potential 有沒有符合原來的邊界條件


![](https://i.imgur.com/hf70EYr.png)
:::spoiler Potential 
![](https://i.imgur.com/feanke2.png) 
:::
:::spoiler Electric Field
![](https://i.imgur.com/NQeVv3e.png)
:::

![](https://i.imgur.com/NCEiXdH.png)
:::spoiler Surface Charge Density
![](https://i.imgur.com/Vmdcw5s.png)
:::
:::spoiler Total Charge on the Conducting Surface
![](https://i.imgur.com/dajZ4fE.png)
:::



## A Line Charge or Parallel Conducting Cylinder above a Grounded Conducting Plane
透過高斯定律可以快速寫出電場
![](https://i.imgur.com/YUCo85W.png) 
:::spoiler 電場
![](https://i.imgur.com/FFLQTWg.png)
:::
:::spoiler 電位
![](https://i.imgur.com/n5tTva9.png) ![](https://i.imgur.com/bXfcSsZ.png)
換成 
![](https://i.imgur.com/XtuQMlO.png)
:::
![](https://i.imgur.com/AE0b5BT.png)

柱狀面，截面是個圓
![](https://i.imgur.com/653xc5G.jpg)
:::spoiler PF
![](https://i.imgur.com/VJwEo0l.png)
:::spoiler 另解
![](https://i.imgur.com/VKQPJcx.png)
:::
Equipotential and electric field for two parallel line charges
![](https://i.imgur.com/lYfPdSt.png)

Determination of axial location and radius of the equipotential surface for two parallel line charges
1. ![](https://i.imgur.com/uw5KHE2.png)
2. ![](https://i.imgur.com/ypTciWe.png)
3. ![](https://i.imgur.com/DA5eDYF.png)
4. ![](https://i.imgur.com/Pv7b5O4.png)

![](https://i.imgur.com/SDMWo32.png)
Equipotential of a cylindrical conductor by a grounded plane
![](https://i.imgur.com/6DpVruq.png)
![](https://i.imgur.com/Q846QOP.png)
![](https://i.imgur.com/r9BIVWY.png)
![](https://i.imgur.com/BrOKHir.png)

:::spoiler Equivalent problems x4
1. Two lines with opposite line charge densities, separated by 2b
2. One line charge by a grounded plane by b
3. One charged cylinder by a grounded plane
4. Two parallel wires, separated by D, with equal but opposite charge, and with the same or different radii
:::
# Capacitance of two charged wires
![](https://i.imgur.com/Fd5YHSe.png)
經過  ![](https://i.imgur.com/OHAYtHi.png) 就可以推導出
![](https://i.imgur.com/fOQkP4T.png)
## two identical wires with opposite line charges
:::danger 
這幾個很相像的公式，建議可以進行比較，方便記憶且加深印象
多看就會有印象
:::
![](https://i.imgur.com/Wcf2vdb.png)
![](https://i.imgur.com/MCyt60U.png)
![](https://i.imgur.com/ahO7NwL.png)

## Off-center parallel conducting wires

![](https://i.imgur.com/KsLvYPC.png)
![](https://i.imgur.com/cSoBhXt.png)
特殊通式
![](https://i.imgur.com/zMA8ltD.png)
:::danger 
這裡的重點是如何得到這些公式，不要去死記，因為我根本也記不住
:::

## A point charge by a grounded spherical conductor
![](https://i.imgur.com/p9x7dsy.png)
![](https://i.imgur.com/CGCLEAg.png)
:::info
剩餘球的部分留在 Weekly Report 2.4 在整理
:::
# 心得
這整段就是過去學過的工數內容的應用，相對沒有新鮮感，這個禮拜的 loading 相當重阿，大部分都是數學的推導，有些推導過程有類似的部分，考試前強烈建議自己重新推推看，而且可以把類似的部分拿出來做比較，透過對照找出不同之處可以讓我們對於該部分的印象特別加深。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熟，知道怎麼來了之後，只記得她的結果的話，那麼就沒有想像中那麼難，寫題目也不需要這樣推，所以應該會比較簡單吧...嗎




















