This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: HW6 1.A rectangular pipe, running parallel to the z axis (from  to +), has three ground metal sides at y=0, y=a, and x=0. The fourth side, at x=b, is maintained at a specific potential V(y). (a)Develop a general formula for the potential within the pipe. (b)Find the potential explicitly for the case of V(y)= V0 (constant). Solution:(a)With the method of separation of variables, the Laplaces equation of 2222yVxVhas the solution of )()(yYxXV. )()()()(222122yYCdyyYdxXCdxxXdwith 21CC. With the boundary condition of V=0 at y=0 and at y=a, we immediately obtain 22sin)(aynCandaynyYn is integer. Thus 21aynC. From V=0 at x=0, the general solution of X is axnxXsinh)(So the general solution is aynaxnAyxVnnsinsinh),(1(b)The coefficient An can be calculated as dyaynybVabnaAansin),(sinh2Then for V(b,y)=V. evennoddnabnnVnabnanaVdyaynabnaVAansinh4cos1sinh2sinsinh22.A cubical box (side length of a) consists of five metal plates (x=0, x=a, y=0, y=a, and z=0), which are welded together and grounded. The top (z=a) is made of a separate sheet of metal, insulated from the others, and held at a constant potential V. Find the potential inside the box. Solution:With the method of separation of variables, the Laplaces equation of 222222zVyVxVhas the solution of )()()(zZyYxXV. From V=0 at x=0 and x=a, the general solution of X is axnxXsin)(n is integer....
View
Full
Document
This note was uploaded on 02/06/2010 for the course PHYSICS 11 taught by Professor Qiu during the Fall '09 term at University of California, Berkeley.
 Fall '09
 Qiu
 Magnetism

Click to edit the document details