Energy Density / Energy Flux / Total Energy in 3D

Transcription

1 Lecture 5 Phys 75 Energy Density / Energy Fux / Tota Energy in D Overview and Motivation: In this ecture we extend the discussion of the energy associated with wave otion to waves described by the D wave euation. In fact the first part of the discussion is exacty the sae as the 1D case just extended to D. In the exapes we oo at the energy associated with sphericay syetric waves. Key Matheatics: oe D cacuus especiay the divergence theore and the spherica-coordinates version of the gradient. I. Density Fux and the Continuity Euation As in the 1D case et's assue that we are interested in soe uantity Q () t that has an associated density ρ ( x. ince we are deaing with a density that ives in a D space ρ = Q. the units of density wi be the units of Q divided by. That is [ ] [ ] Let's consider a voue encosed by a surface as iustrated in the foowing figure. At each point on the surface we define a perpendicuar outward-pointing unit vector nˆ ( r ) associated with each point r on the surface. r nˆ nˆ ( r ) nˆ ( r ) encosed voue The aount of Q contained in can be written as () t ρ ( r d (1) Q = r As in the 1D case if Q is a conserved uantity then the change in Q inside D M Riffe -1- /1/1

2 Lecture 5 Phys 75 () t ρ( r dq = d r () dt ust be eua to the net fow of Q into dq dt () t ( r nˆ ( r ) d. () = j The (vector) uantity j is again nown as the Q current density or the Q fux. The diensions of j are the diensions of ρ ties a veocity so [ j ] = [ ρ] s. Thus aso [] j = [ Q]( s). Note that the rhs of E. () can be interpreted as the tota Q current fowing out through the surface. Euating the rhs's of Es. () and () gives us ρ ( r d r = j ( r nˆ ( r) d. (4) We can now use the divergence theore (which is one of severa D extensions of the fundaenta theore of cacuus) A ( r ) d r = A( r) nˆ ( r ) d (5) to rewrite E. (4) as ρ ( r + j ( r d r = (6) Now because the voue is arbitrary the integrand ust vanish. Thus ρ ( r + j ( r =. (7) Euation (7) is the D version of the continuity euation which again is a oca stateent of the conservation of Q. D M Riffe -- /1/1

3 Lecture 5 Phys 75 II. Energy Density and Fux for D Waves We now appy this discussion to the energy associated with D waves. In this case Q represents the energy associated with a wave (within soe voue). For waves described by the D wave euation the energy density can be written as µ ρ( r = + c ( ) (8) where ( r is the variabe that is governed by the wave euation. The first ter on the rhs of E. (8) is the inetic energy density ρ T and the second is the potentia energy density ρ V. Now E. (8) is fairy genera as ong as µ is suitaby interpreted. If is a true dispaceent then µ wi be a paraeter with the units of ass density. If represents soething ese say an eectric fied then it wi have soe other units. Fro E. (8) it is fairy easy to see that the units of µ are generay given by [ µ ]= (Joue s )/( [] ). It is not hard to show that the energy fux which can be written as j ( r = µ c (9) together with the energy density in E. (8) satisfy E. (7) the continuity euation. III. evera Exapes Let's oo at soe exapes that invove sphericay syetric waves. A. pherica tanding Wave Let's oo at a standing-wave exape. You ay reca that a spherica-coordinates separabe soution that is finite everywhere is of the for iφ iφ ict ict ( r θ φ = C j ( ) P ( cos( θ )) ( C e + D e ) ( A e + B e ) (1) where j is a spherica Besse function (of the first ind) and P is an associated Legendre function (of the first ind). The paraeter is an integer whose absoute vaue can be no arger than the nonnegative integer. If we want a soution with spherica syetry then there can be no θ or φ dependence. This eans that both and ust be zero because the ony associated Legendre function independent of θ is P ( cos( θ )) = 1. Thus the spherica Besse function in E. (1) ust be j ( ) = sin( ) ( ) and so E. (1) sipifies to D M Riffe -- /1/1

4 Lecture 5 Phys 75 sin ( ) ( ict ict r θ t = C A e + B e ). (11) φ If we sipify this further by etting A be a rea nuber and et soution expicity rea) then we have B = A (aing the sin ( ) r θ φ t A C cos( c. (1) = Because a parts of the syste osciate with the sae phase this is a sphericay syetric version of a standing wave. Using Es. (8) and (9) we can cacuate the inetic and potentia energy densities and the energy fux associated with the wave in E. (1). To do this in a fairy sipe anner we can use the spherica-coordinates version of the gradient f 1 f 1 f f ( r θ φ) = rˆ + θˆ + φˆ (1) r r θ r sin( θ ) φ where rˆ θˆ and φˆ are unit vectors in the r θ and φ directions respectivey. The nice thing about sphericay syetric soutions is that ony the first ter on the rhs of E. (1) contributes to the gradient. A video of ρ T ρ V and j for the wave in E. (1) Energy in D tanding Wave.avi is avaiabe on the cass web site. As the video shows the dispaceent is indeed a standing wave. Unfortunatey the energy densities and fux fa off with the radia distance r so fast that it is hard to reay see their behavior. Given this we have ade another video Energy in D tanding Wave.avi which pots the surface integrated density and fux 1 and D I () r ρ ( r ) d (14) = () r = j() r nˆ ( r ) d (15) 1 The video separatey shows the inetic and potentia contributions to D ( r). D M Riffe -4- /1/1

5 Lecture 5 Phys 75 where the surface is of radius r centered at the origin. Now because a sphericay syetric soution is independent of the two anges θ and φ this aounts to utipying the density ρ and fux j by the factor 4π r which is the surface area of the sphere. The uantity D () r (which has units of Joue/) can be though of as a inear energy density (i.e. the energy per unit ength aong the radia direction) whie the uantity I () r (which has units of Joue/s) is the tota (energy) current fowing through. Notice that this new video is very siiar to the 1D standing wave video that we ooed at in the ast ecture. B. pherica Traveing Wave Let's aso oo at a sphericay syetric traveing wave. If we are thining about sound waves this is the sort of wave that woud resut fro a pusating sphere centered at the origin. We can construct a traveing wave soution fro a inear cobination of ineary independent standing waves. We thus need to use both inds of spherica Besse functions. The inear cobination that produces a sphericay syetric outgoing traveing wave is ( r θ φ = C [ j ( ) cos( c y ( ) sin( c ]. (16) which can be written in ters of sine and cosine functions as sin ( ) cos ( ) r θ t = C cos ct sin( c. (17) φ The video Energy in D Traveing Wave.avi shows D ( r) and I ( r) for this wave. Indeed away fro the origin the wave appears to be an outgoing traveing wave. However at the origin soething rather different sees to be happening soething with soe standing-wave character perhaps? We as it turns out the current density j has ters with two types of behavior. The first type has a 1 r dependence. These ters describe the radiative part of the wave which carries energy off to infinity. Because the radiative part of j varies as 1 r the radiative part of I () r does not vanish as r. However there are aso nonradiative ters which vary as 1 r. These ters act ore ie a standing wave: the energy associated with these ters just osciates bac and forth and never reay goes anywhere. Because of the 1 r behavior to the nonradiative part of j the current I () r associated with these ters vanishes as 1 r as r. These ters are Note that this soution is ony vaid in the region of space outside the source. In the video you ay thin of the source as being infinitesiay sa so that the soution is vaid infinitesiay cose to the origin. D M Riffe -5- /1/1

6 Lecture 5 Phys 75 thus soeties caed the oca fieds associated with the source. In the video Energy in D Traveing Wave.avi we separatey show the current I ( r) associated with each type of ter. Notice that the radiative piece oos essentiay ie a 1D traveing wave whie the nonradiative piece is reay ony iportant in the vicinity of the origin. Exercises *5.1 how that the expressions for the density ρ and j in Es. (8) and (9) respectivey satisfy E. (7) the continuity euation. **5. pherica Traveing Wave (a) Write the wave in E. (17) ( r ( ) ( ) sin cos θ = C cos( c sin( c φ as an expicit function of ( r c thus showing that it is a traveing wave oving outward fro the origin. (b) Using your resut fro part (a) show that the radiative and nonradiative coponents of the current density j can be written respectivey as j R c ( r = µ cos ( crˆ and j ( r cos( c sin( crˆ r NR c = µ. (c) Cacuate the tie average of each of these current-density coponents (defined as 1 T j ( r dt where T is one period of osciation) and show that the average of the T radiative part points in the positive rˆ direction whie the tie average of the nonradiative part is zero. (Note: neither answer shoud have any dependence on T.) *5. Pane Wave Energy Density. Consider the pane-wave soution to the D wave euation ( x y z = exp{ i( xx + y y + z z c }. (a) Cacuate the inetic potentia and tota energy densities ρ T ( x y z ρ V ( x y z and ρ ( x y z respectivey and the energy current density j ( x y z. (b) how that the D continuity euation is satisfied by your expressions for ρ and j. D M Riffe -6- /1/1

7 Lecture 5 Phys 75 **5.4 pherica tanding Wave Energy Density. Consider the sphericay syetric standing wave soution to the D wave euation sin ( ) r θ φ = cos( c. (a) Cacuate the inetic potentia and tota energy densities ρ T ( r θ φ ρ V ( r θ φ and ρ ( r θ φ respectivey. (b) how for arge distances fro the origin ( >> 1) that the tota energy density for µ this wave is approxiatey sin cos ρ r θ φ t = c sin ct + cos( c. D M Riffe -7- /1/1