MIT6_641s09_sol_exam2009

MIT6_641s09_sol_exam2009 - MIT OpenCourseWare...

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MIT OpenCourseWare http://ocw.mit.edu 6.641 Electromagnetic Fields, Forces, and Motion Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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± ± ± ± ± ± ± ± 6.641 Electromagnetic Fields, Forces, and Motion Spring 2009 Final - Solutions - Spring 2009 Prof. Markus Zahn MIT OpenCourseWare Problem 1 Figure 1: A diagram of a sheet of surface charge at y = 0 between two grounded perfect conductors at y = b and y = a (Image by MIT OpenCourseWare). A sheet of surface charge with surface charge distribution σ s ( x,y = 0) = σ 0 sin kx is placed at y = 0, parallel and between two parallel grounded perfect conductors at zero potential at y = b and y = a . The regions above and below the potential sheet have dielectric permittivities of 2 and 1 . Neglect fringing ±eld effects. A Question: What are the electric potential solutions in the regions 0 y a and b y 0 ? Solution: A sinh k ( y a )sin kx 0 < y < a Φ( x,y ) = B sinh k ( y + b )sin kx b < y < 0 Φ( x,y = 0 ) = Φ( x,y = 0 + ) ⇒ − A sinh ka = B sinh kb Φ E y ( x,y = 0 + ) = = Ak cosh k ( y a )sin kx = Ak cosh ka sin kx y =0+ y =0 + ∂y Φ E y ( x,y = 0 ) = = Bk cosh k ( y + b )sin kx = Bk cosh kb sin kx y =0 y =0 ∂y 1
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± ± ± C Final - Solutions - Spring 2009 6.641, Spring 2009 σ s ( x,y = 0) = σ 0 sin kx = 2 E y ( x,y = 0 + ) 1 E y ( x,y = 0 ) = 2 Ak cosh ka sin kx + 1 Bk cosh kb sin kx σ 0 = 2 Ak cosh ka + 1 Bk cosh kb A sinh ka A sinh ka cosh kb B = sinh kb ⇒ − 2 Ak cosh ka 1 k sinh kb = σ 0 ⇒ − Ak 2 cosh ka + 1 sinh ka cosh kb = σ 0 sinh kb σ 0 sinh kb A = k [ 2 cosh ka sinh kb + 1 sinh ka cosh kb ] A sinh ka σ 0 sinh ka B = = sinh kb k [ 2 cosh ka sinh kb + 1 sinh ka cosh kb ] Φ( x,y ) = σ 0 sinh kb sinh k ( y a )sin kx A sinh k ( y a )sin kx = k [ 2 cosh ka sinh kb + 1 sinh ka cosh kb ] 0 < y < a σ 0 sinh ka sinh k ( y + b )sin kx B sinh k ( y + b )sin kx = k [ 2 cosh ka sinh kb + 1 sinh ka cosh kb ] b < y < 0 B Question: What are the electric feld distributions in the regions 0 < y < a and b < y < 0 ? Solution:
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This note was uploaded on 12/02/2010 for the course ECE 6.641 taught by Professor Zahn during the Spring '09 term at MIT.

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MIT6_641s09_sol_exam2009 - MIT OpenCourseWare...

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