# 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