Problem.Set.5.S09b

Problem.Set.5.S09b - 3 Inside a grounded hollow right...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
411 Problem Set 5 2009 1 . Using the solution to Laplace’s equation in cylindrical coordinates, 2 ( , , z )= 0, where ( , , z )= , m [ a m J m (  )+ b m N m (  )] e ± im e ± z write a general form for the Green’s function, 1 4 r r , in cylindrical coordinates for all of space where 2 G ( r , r ) = ( r r ) and G ( r , r ) 0for | r |→ ∞ , r fixed and | r |→ ∞ , r fixed. Incorporate the discontinuity in the derivative of G ( r , r ) in and assume that z remains finite. [Note that one can use Hankel functions, H m ± (  )≡ J m (  iN m (  ) , rather than J m (  ) and N m (  ) .If is pure imaginary ( ik ) the dependence is given by the modified Bessel functions, I m (  )= i m J m ( i  ) and K m (  )= 2 i m + 1 H m + ( i  ) This permits use of e ikz ] 2. A charge distribution is given by ( r )= Q a 5 xy Θ ( a r ) . Using the Green’s function formalism and the Green’s function for all of space, find the electrostatic potential, ( r ) ,for all r satisfying. ( r )→ 1 r as r →∞ . Show all the steps leading to the final expression.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 3. Inside a grounded hollow right circular cylinder of radius a with symmetry axis along z axis and ends at z = 0 and L the electrostatic potential,  (  ,  , z ) , satisfies ∇ 2  (  ,  , z ) = − 4  Q a 3  (  − b ) Θ ( 2 L / 3 − z ) Θ ( z − L / 3 ) Using the Green’s function satisfying ∇ 2 G D ( r , r ′ ) =  ( r − r ′ ) and G D ( r , r ′ ) = 0 on the cylinder find the electrostatic potential in Gaussian units for all points inside the cylinder. The G D ( r , r ′ ) can be obtained from the following table using the conversion −  q  ( x , x ′ ) = G D ( r , r ′ ) .While any of the forms in the table could be used, for this problem choose the first expression. It is not necessary to derive the Green’s function or to determine its expansion coefficient....
View Full Document

This note was uploaded on 12/19/2010 for the course PHYS 411 taught by Professor G during the Spring '10 term at Missouri S&T.

Page1 / 2

Problem.Set.5.S09b - 3 Inside a grounded hollow right...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online