lecture8 - Lecture 8 Boundary Value Problems In this...

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1 ECE 303 – Fall 2007 – Farhan Rana – Cornell University Lecture 8 Boundary Value Problems In this lecture you will learn: • How to solve some interesting boundary value problems ECE 303 – Fall 2007 – Farhan Rana – Cornell University Solutions of Laplace Equation in Spherical Coordinates () B r A r + = r φ Spherically Symmetric Solution A Constant Uniform Electric Field Solution ( ) () z A r A r = = θ cos r z () () z A r r E ˆ = −∇ = r r r A Dipole Oriented Along z-Axis Solution 2 cos r A r = r z + q - q For all the following solutions ( ) 0 2 = r r + q
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2 ECE 303 – Fall 2007 – Farhan Rana – Cornell University A Non-Uniformly Charged Spherical Shell - I Consider a charged spherical sheet where the surface charge density is fixed and is given by: θ z () σ cos o = + + + + + ++ - -- Looking from outside the potential sort of looks like that of a dipole. So try a dipole-like solution for r > a a ( ) C r A r out + = 2 cos φ r For r > a : For r < a : Try something that does not go to infinity at r = 0 ( ) ( ) D r B r in + = cos r 0 0 Why?? Why?? ECE 303 – Fall 2007 – Farhan Rana – Cornell University A Non-Uniformly Charged Spherical Shell - II z + + + + + a How would you know if your solution is the right one?? Uniqueness Theorem: There is only one right solution and If you have a solution that satisfies all the boundary conditions then you have the right solution x ( ) 0 . 0 0 = = = = = r s d E r r r r r r r s d r For r < a : ( ) ( ) D r B r in + = cos r 0 Why??
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3 ECE 303 – Fall 2007 – Farhan Rana – Cornell University A Non-Uniformly Charged Spherical Shell - III θ z + + + + + ++ - -- a () σ cos o = ( ) 2 cos r A r out φ = r ( ) ( ) cos r B r in = r Boundary Conditions: (1) Potential inside and outside must be continuous at the surface of the shell ( ) ( ) cos cos 2 a B a A r r a r out a r in = = = = r r ( ) o o o a r out o a r out o a r r in a r r out o B a A r r r r E E ε cos cos cos 2 cos 3 , , = + = + = = = = = r r (2) Discontinuity in the radial electric field at the surface must be related to the local surface charge density ECE 303 – Fall 2007 – Farhan Rana – Cornell University A Non-Uniformly Charged Spherical Shell - IV z + + + + + cos o = A surface charge density of the form: produces a uniform z-directed E-field inside the charged shell and a dipole-like E-field outside the charged shell Solution: ( ) 2 3 cos 3 r a r o o out
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This note was uploaded on 02/02/2008 for the course ECE 3030 taught by Professor Rana during the Fall '06 term at Cornell.

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lecture8 - Lecture 8 Boundary Value Problems In this...

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