Problem.Set.2.soln.2009

Problem.Set.2.soln.2009 - Problem Set 2: 2.1 The Coulomb...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Problem Set 2 : 2 . 1 The Coulomb potential for a point charge located at r 0 is given by ( r , r 0 ) = q | r r 0 | a) Find 2 ( r , r 0 ) and write down the partial differential equation satisfied by ( r , r 0 ) . b) Using steps similar to those found on page 12 of Chapter 1 for the wave equation, find the fourier transform, ( k , r 0 ), of the Coulomb potential. c) Find ( k , r 0 ) directly from ( r , r 0 ) = 1 ( 2 ) 3 ∫∫∫ ( k , r 0 ) exp ( i k r ) d 3 k d) Evaluate the following integral 0 sin ( ku ) du Solution: a) 2 ( r , r 0 )=∇ 2 q | r r 0 | =− 4 q ( r r 0 ) so ( r , r 0 ) satisfies 2 ( r , r 0 )=− 4 q ( r r 0 ) b) One can take the Fourier transform of this equation: 2 1 ( 2 ) 3 ∫∫∫ ( k , r 0 ) exp ( i k r ) d 3 k =− 4 q ( 2 ) 3 ∫∫∫ exp ( i k ( r r 0 )) d 3 k ∫∫∫ [ ( k , r 0 )(− k k )+ 4 q exp (− i k r 0 )] exp i k r ) d 3 k = 0 Since the exp i k r are linearly independent functions for each k , ( k , r 0 )(− k k )+ 4 q exp (− i k r 0 )= 0 and ( k , r 0 ) = 4 q exp (− i k r 0 ) k k c) The direct method for finding ( k , r 0 ) is as follows.: ( r , r 0 ) = q | r r 0 | = 1 ( 2 ) 3 ∫∫∫ ( k , r 0 ) exp ( i k r ) d 3 k and
Background image of page 1

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

View Full DocumentRight Arrow Icon
( k , r 0 ) = ∫∫∫ q | r r 0 | exp (− i k r ) d 3 r = q exp (− i k r 0 ) ∫∫∫ exp (− i k u ) u d 3 u ; u = r r 0 = q exp (− i k r 0 ) 0 0 2 0 exp (− iku cos ) u u 2 du sin d d = 2 q exp (− i k r 0 ) 0 u 1 + 1 exp (− iku ) d du ; = cos = 2 q exp (− i k r 0 ) 0 u exp (− iku ) − exp ( iku ) iku du = 4 q k exp (− i k r 0 ) 0 sin ( ku ) du d) By comparison with the result in c): 0 sin ( ku ) du = 1 k
Background image of page 2
Assignment 3 : The ‘displacement’ of a damped, harmonic oscillator satisfies the equation x ̈ + x ̇ + 0 2 x = f ( t ) . (a) Obtain the Green’s function for this equation. That is, find g ( t t ) such that g ̈ ( t t ) + g ́ ( t t ) + 0 2 g ( t t ) = ( t t ) (b) In the case that the ‘force’ is given by
Background image of page 3

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

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/19/2010 for the course PHYS 423 taught by Professor G. during the Spring '10 term at Missouri S&T.

Page1 / 8

Problem.Set.2.soln.2009 - Problem Set 2: 2.1 The Coulomb...

This preview shows document pages 1 - 4. Sign up to view the full document.

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