1. 講者： 許永昌 老師 1

2. Contents Definition Other forms Recurrence Relation Wronskian Formulas Example: Coaxial Wave Guides 2

3. Definition ( 請預讀 P699~P700) Neumann Functions: Why do we need to define a new solution for Bessel Eq.? When , J and J - are independent to each other; however, J -n =(-1) n J n when n  . We need to find a second solution. (Ch 9.6) The asymptotic series of J (z)~ we want to find a solution N (z)~ 3

4. Series Form ( 請預讀 P700) 1) 4 Reference: http://mathworld.wolfram.com/BesselFunctionoftheSecondKind.htmlhttp://mathworld.wolfram.com/BesselFunctionoftheSecondKind.html Q: How about N ?

5. Other Forms ( 請預讀 P701) From the contour integral representation of Hankel function 5

6. Recurrence relations ( 請預讀 P702) 6

7. Wronskian Formulas ( 請預讀 P702) Wronskian (See Ch9.6): Therefore, Find A from x  0. 7

8. Wronskian Formulas (continue) 8

9. Wronskian Formulas (continue) 9

10. Example ( 請預讀 P703~P704) Coaxial Wave Guide: Conditions: For TM mode (H z =0 everywhere)  z E=ikE, Under these two conditions, we can get E z =0 on the boundary ( Faraday’s Law) Therefore, we just consider the PDE for E z. 10

11. Example (continue) PDE:  2 E z =(  /c) 2 E z. Separation Variables: Boundary condition: E z (a)= E z (b)=0 11 Therefore,  is quantized. If  and  are provided, Therefore,  is quantized. If  and  are provided,

12. Example (for TE mode) (continue) Boundary conditions: n  E=0 on the boundary (Faraday’s Law) B  =0 on the boundary (Gauss Law of magnetic field or  E=i  B.) For TE mode: E z =0 everywhere   H z =0 on the boundary (Ampere’s Law). PDE:  2 H z =(  /c) 2 H z. 12

13. Example (for TEM mode) (continue) For TEM mode: E z =B z =0 everywhere. If we use  =  t +ê 3  z for Maxwell Eqs., we will get Therefore, there is no TEM mode for a hollow cylindrical wave guide, but TEM mode can be survived in a coaxial cylindrical wave guide. 13

14. Homework 11.3.2 (12.2.2e) 11.3.3 (12.2.3e) 14

15. Nouns 15