I need help calculating the impact speed of a positron and electron starting at 5 cm apart
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From: Timo Nieminen on 10 May 2010 16:12 On May 10, 2:38 pm, franklinhu <frankli...(a)yahoo.com> wrote: > Let's say we have a positron and electron at rest with respect to each > other. Since they are oppositely charged, they are immediately > attracted to each other and begin to accelerate toward each other. At > any point in the path, you can calculate the force between them using > Columbs law, based on that, you could calculate the acceleration, but > how do you calculate what would be the velocity of the positron and > electron as they approach each other and then collide from some > starting distance like 5cm? As distance approaches zero, the force > approaches infinity. Does this mean that the velocity may approach > infinity due to the infinite force acting on a mass or is there some > limiting mechanism? Please help? (1) The classical physics of charged point particles is broken. (2) If they were charged point particles of mass m_e (i.e., the electron mass), then the velocity would approach infinity. PE -> KE and all that. (3) A classical charged point particle, in the absence of renormalisation, has infinite inertia so they'd never move towards each; they'd just sit there with their infinite masses. (Yes, a theoretical result, but surely this is OK since classical point particles are purely theoretical entities.) (4) What do you get for a classical electron, of the classical electron radius, colliding with a classical positron, also of the classical electron radius? Don't start at 5cm away; start an infinite distance away and use PE -> KE. Ignore radiative reaction. (5) When theory fails, resort to the real world. What happens to the kinetic energy of an electron and positron when they annihilate? Compare the energy of the emitted gammas to the initial rest energy. The difference is what the KE was. How does this compare with the result from (4). It's certainly much less than the result from (2).
From: franklinhu on 12 May 2010 01:44 On May 10, 1:12 pm, Timo Nieminen <t...(a)physics.uq.edu.au> wrote: > On May 10, 2:38 pm, franklinhu <frankli...(a)yahoo.com> wrote: > > > Let's say we have a positron and electron at rest with respect to each > > other. Since they are oppositely charged, they are immediately > > attracted to each other and begin to accelerate toward each other. At > > any point in the path, you can calculate the force between them using > > Columbs law, based on that, you could calculate the acceleration, but > > how do you calculate what would be the velocity of the positron and > > electron as they approach each other and then collide from some > > starting distance like 5cm? As distance approaches zero, the force > > approaches infinity. Does this mean that the velocity may approach > > infinity due to the infinite force acting on a mass or is there some > > limiting mechanism? Please help? > > (1) The classical physics of charged point particles is broken. What do you mean by 'broken'. Do you mean that they don't apply due to unknown reasons or because they simply don't apply for a specific reason. Or do you mean that the physics in this area is generally broken and you cannot solve this problem at all using all that modern physics has avaliable. I would find that hard to believe since the positron / electron collision must have been studied to death. > > (2) If they were charged point particles of mass m_e (i.e., the > electron mass), then the velocity would approach infinity. PE -> KE > and all that. Really???? You are saying that theoretically, the velocity would indeed approach infinity if there are no other limits? I tried doing a calculation, and as you get closer, the amount you can acclerate decreases. It seemed accleration was limited by the initial starting distance, but that didn't make any sense either, so I think I got the calculation wrong. However, I think that we could safely put a cap on the speed at being the speed of light. Nothing should go faster than that. So if what you are saying is correct, we could presume that at some point, the electron and positron will attain the speed of light at some point and then go no faster. > > (3) A classical charged point particle, in the absence of > renormalisation, has infinite inertia so they'd never move towards > each; they'd just sit there with their infinite masses. (Yes, a > theoretical result, but surely this is OK since classical point > particles are purely theoretical entities.) > Yes, this is the point made by other posters in that if there is no limit to how small an increment you could consider, then the positron/ electron would never reach eachother. But, we know that they do eventually reach each other to annihillate, so this is not a reasonable conclusion. > (4) What do you get for a classical electron, of the classical > electron radius, colliding with a classical positron, also of the > classical electron radius? Don't start at 5cm away; start an infinite > distance away and use PE -> KE. Ignore radiative reaction. > Well if you plug into the PE formula KQq/r where r= 2.8E-15, you get 8.22E-14 Joules Curiously, this is nearly identical to the E=mc^2 formula which works out to 8.19E-14 Joules > (5) When theory fails, resort to the real world. What happens to the > kinetic energy of an electron and positron when they annihilate? > Compare the energy of the emitted gammas to the initial rest energy. > The difference is what the KE was. How does this compare with the > result from (4). It's certainly much less than the result from (2). OK, we know that in the real world, the energy of the emitted gammas is 8.19E-14 Joules. We know that the calculated kinetic energy of collsion should be around 8.22E-14 Joules. Now if this was the collision of anything other than a positron and electron, lets say we have 2 blobs of clay with the mass of an electron that have an inelastic collision, we would expect that the collison would release 8.22E-14 Joules of energy based upon the potential energy. We could also back calculate what the speed of the collsion should be since we divided the avalaible KE energy by 2 and then figure out v from KE=1/2mv^2. We would find that the particle speed prior to collison would need to be close to 3E8 m/s which we would recognized as the speed of light. So, we would naturally think that the 8.22E-14 joules of energy released from the collsion was simply the release of the kinetic energy from the 2 blobs of clay colliding at the speed of light. But wait, we aren't talking about blobs of clay, we are talking about electrons and positrons - and because of that, we think that the 8.19E-14 joules of energy released is not due to the electron and positron colliding with each other, but is rather due to the mysterious conversion of matter to energy in accordance to E=mc^2. But wait again, we just calculated that if we had a positron and electron, that if they did collide in an inelastic collision at an electron radius, that it should release 8.22E-14 joules of energy. So really, we should see close to 16E-14 joules of energy. 8E-14 from the conversion of matter to energy plus an additional 8E-14 from the release of the kinetic energy from the collision. But of course, we don't see 16E-14 joules of energy. So what is the answer to this mystery.... I'll give you a moment to think about it... I don't know about you, but I would think the logical conclusion would be that what we think is the energy released in the annihillation event, is really just the release of the kinetic energy from the collision of the positron and the electron. Before you go nahhhhh, can't be ... think about it. We just made a legitimate calculation of the kinetic energy of a positron/electron collision. It just happens to be by pure coincidence, the same as the E=mc^2 formula eventhough the format and numbers of the PE formula and E=mc^2 forumulas look nothing alike and you'd think there would be no relationship. I just made that calculation while writing this post, and there can be no better 'AH-HA!!' moment than that to see both numbers line up. Of course, you would have to abandon this notion that matter 'converts' into energy since matter is conserved absolutely in classical inelastic collisions. This would mean that the positron and electron remain after the collision in a form which is typically undetectable. But that is the topic of another post. Thanks for the tips Timo!
From: Timo Nieminen on 12 May 2010 05:21 On May 12, 3:44 pm, franklinhu <frankli...(a)yahoo.com> wrote: > On May 10, 1:12 pm, Timo Nieminen <t...(a)physics.uq.edu.au> wrote: > > On May 10, 2:38 pm, franklinhu <frankli...(a)yahoo.com> wrote: > > > > Let's say we have a positron and electron at rest with respect to each > > > other. Since they are oppositely charged, they are immediately > > > attracted to each other and begin to accelerate toward each other. At > > > any point in the path, you can calculate the force between them using > > > Columbs law, based on that, you could calculate the acceleration, but > > > how do you calculate what would be the velocity of the positron and > > > electron as they approach each other and then collide from some > > > starting distance like 5cm? As distance approaches zero, the force > > > approaches infinity. Does this mean that the velocity may approach > > > infinity due to the infinite force acting on a mass or is there some > > > limiting mechanism? Please help? > > > (1) The classical physics of charged point particles is broken. > > What do you mean by 'broken'. Do you mean that they don't apply due to > unknown reasons or because they simply don't apply for a specific > reason. Or do you mean that the physics in this area is generally > broken and you cannot solve this problem at all using all that modern > physics has avaliable. I would find that hard to believe since the > positron / electron collision must have been studied to death. Basically, point (3) below. A classical point charge has infinite energy in its field, and infinite inertia as a consequence. That isn't quite the whole problem, but it's a central problem. See M. Frisch, "Inconsistency, asymmetry, and non-locality: A philosophical investigation of classical electrodynamics", OUP 2005. (For a good look at what you can do in practice despite this - if you're happy with renormalisation, see F. Rohrlich, "Classical charged particles", 3E, World Scientific 2007.) > > (2) If they were charged point particles of mass m_e (i.e., the > > electron mass), then the velocity would approach infinity. PE -> KE > > and all that. > > Really???? You are saying that theoretically, the velocity would > indeed approach infinity if there are no other limits? I tried doing a > calculation, and as you get closer, the amount you can acclerate > decreases. It seemed accleration was limited by the initial starting > distance, but that didn't make any sense either, so I think I got the > calculation wrong. > > However, I think that we could safely put a cap on the speed at being > the speed of light. Nothing should go faster than that. So if what you > are saying is correct, we could presume that at some point, the > electron and positron will attain the speed of light at some point and > then go no faster. With classical mechanics, you get speed -> infinity. With relativistic mechanics, you have speed -> c. Either way, infinite kinetic energy. Sounds like your attempted calculation was wrong. For an inverse- square force, the KE goes to infinity. For other force laws, you can get infinity, or some finite amount (which will depend on the starting distance). > > (3) A classical charged point particle, in the absence of > > renormalisation, has infinite inertia so they'd never move towards > > each; they'd just sit there with their infinite masses. (Yes, a > > theoretical result, but surely this is OK since classical point > > particles are purely theoretical entities.) > > Yes, this is the point made by other posters in that if there is no > limit to how small an increment you could consider, then the positron/ > electron would never reach eachother. No, this isn't that same point at all. If they have infinite inertia, they never accelerate towards each other at all, if the force is short of infinite. This isn't what we see. Worse, the observed upper limit to the size of the electron means a classical mass much greater than what we observe. Thus, point (1). > But, we know that they do > eventually reach each other to annihillate, so this is not a > reasonable conclusion. > > > (4) What do you get for a classical electron, of the classical > > electron radius, colliding with a classical positron, also of the > > classical electron radius? Don't start at 5cm away; start an infinite > > distance away and use PE -> KE. Ignore radiative reaction. > > Well if you plug into the PE formula KQq/r where r= 2.8E-15, you get > 8.22E-14 Joules > > Curiously, this is nearly identical to the E=mc^2 formula which works > out to 8.19E-14 Joules Not a coincidence. The classical electron radius is the radius that gives you the energy that gives you the observed inertia. We know the electron is smaller than this. Again, see point (1) above. When you get to near this point, you need to be very careful when trying to use any classical result for point electrons. The safe approach is to not trust classical results. (As a technical nitpick, note that if you do this calculation for an electron-positron pair, they can only approach to within 2 r_e before they hit.) > > (5) When theory fails, resort to the real world. What happens to the > > kinetic energy of an electron and positron when they annihilate? > > Compare the energy of the emitted gammas to the initial rest energy. > > The difference is what the KE was. How does this compare with the > > result from (4). It's certainly much less than the result from (2). > > OK, we know that in the real world, the energy of the emitted gammas > is 8.19E-14 Joules. > > We know that the calculated kinetic energy of collsion should be > around 8.22E-14 Joules. No, we know that at this point, our classical calculations are wrong. Don't depend on calculations that we _know_ are wrong. Use measurements. Observed energy of annihilation gammas - rest energy of electron and positron = KE. Want higher energy photons? Just use higher speed (therefore higher KE) electrons and positrons. [cut] > So what is the answer to this mystery.... I'll give you a moment to > think about it... > > I don't know about you, but I would think the logical conclusion would > be that what we think is the energy released in the annihillation > event, is really just the release of the kinetic energy from the > collision of the positron and the electron. > > Before you go nahhhhh, can't be ... think about it. We just made a > legitimate calculation of the kinetic energy of a positron/electron > collision. No, we know it isn't legitimate. We know it's wrong. > It just happens to be by pure coincidence, the same as the > E=mc^2 formula eventhough the format and numbers of the PE formula and > E=mc^2 forumulas look nothing alike and you'd think there would be no > relationship. I just made that calculation while writing this post, > and there can be no better 'AH-HA!!' moment than that to see both > numbers line up. No pure coincidence at all, the classical radius is chosen to achieve this. (See http://en.wikipedia.org/wiki/Classical_electron_radius for some details.) Don't get too carried away by this "coincidence". Remember (a) measurements work, so use them to find the KE before collision, and (b) the electron is not a billiard ball with a radius equal to the classical electron radius.
From: Timo Nieminen on 12 May 2010 05:21 On May 12, 3:44 pm, franklinhu <frankli...(a)yahoo.com> wrote: > On May 10, 1:12 pm, Timo Nieminen <t...(a)physics.uq.edu.au> wrote: > > On May 10, 2:38 pm, franklinhu <frankli...(a)yahoo.com> wrote: > > > > Let's say we have a positron and electron at rest with respect to each > > > other. Since they are oppositely charged, they are immediately > > > attracted to each other and begin to accelerate toward each other. At > > > any point in the path, you can calculate the force between them using > > > Columbs law, based on that, you could calculate the acceleration, but > > > how do you calculate what would be the velocity of the positron and > > > electron as they approach each other and then collide from some > > > starting distance like 5cm? As distance approaches zero, the force > > > approaches infinity. Does this mean that the velocity may approach > > > infinity due to the infinite force acting on a mass or is there some > > > limiting mechanism? Please help? > > > (1) The classical physics of charged point particles is broken. > > What do you mean by 'broken'. Do you mean that they don't apply due to > unknown reasons or because they simply don't apply for a specific > reason. Or do you mean that the physics in this area is generally > broken and you cannot solve this problem at all using all that modern > physics has avaliable. I would find that hard to believe since the > positron / electron collision must have been studied to death. Basically, point (3) below. A classical point charge has infinite energy in its field, and infinite inertia as a consequence. That isn't quite the whole problem, but it's a central problem. See M. Frisch, "Inconsistency, asymmetry, and non-locality: A philosophical investigation of classical electrodynamics", OUP 2005. (For a good look at what you can do in practice despite this - if you're happy with renormalisation, see F. Rohrlich, "Classical charged particles", 3E, World Scientific 2007.) > > (2) If they were charged point particles of mass m_e (i.e., the > > electron mass), then the velocity would approach infinity. PE -> KE > > and all that. > > Really???? You are saying that theoretically, the velocity would > indeed approach infinity if there are no other limits? I tried doing a > calculation, and as you get closer, the amount you can acclerate > decreases. It seemed accleration was limited by the initial starting > distance, but that didn't make any sense either, so I think I got the > calculation wrong. > > However, I think that we could safely put a cap on the speed at being > the speed of light. Nothing should go faster than that. So if what you > are saying is correct, we could presume that at some point, the > electron and positron will attain the speed of light at some point and > then go no faster. With classical mechanics, you get speed -> infinity. With relativistic mechanics, you have speed -> c. Either way, infinite kinetic energy. Sounds like your attempted calculation was wrong. For an inverse- square force, the KE goes to infinity. For other force laws, you can get infinity, or some finite amount (which will depend on the starting distance). > > (3) A classical charged point particle, in the absence of > > renormalisation, has infinite inertia so they'd never move towards > > each; they'd just sit there with their infinite masses. (Yes, a > > theoretical result, but surely this is OK since classical point > > particles are purely theoretical entities.) > > Yes, this is the point made by other posters in that if there is no > limit to how small an increment you could consider, then the positron/ > electron would never reach eachother. No, this isn't that same point at all. If they have infinite inertia, they never accelerate towards each other at all, if the force is short of infinite. This isn't what we see. Worse, the observed upper limit to the size of the electron means a classical mass much greater than what we observe. Thus, point (1). > But, we know that they do > eventually reach each other to annihillate, so this is not a > reasonable conclusion. > > > (4) What do you get for a classical electron, of the classical > > electron radius, colliding with a classical positron, also of the > > classical electron radius? Don't start at 5cm away; start an infinite > > distance away and use PE -> KE. Ignore radiative reaction. > > Well if you plug into the PE formula KQq/r where r= 2.8E-15, you get > 8.22E-14 Joules > > Curiously, this is nearly identical to the E=mc^2 formula which works > out to 8.19E-14 Joules Not a coincidence. The classical electron radius is the radius that gives you the energy that gives you the observed inertia. We know the electron is smaller than this. Again, see point (1) above. When you get to near this point, you need to be very careful when trying to use any classical result for point electrons. The safe approach is to not trust classical results. (As a technical nitpick, note that if you do this calculation for an electron-positron pair, they can only approach to within 2 r_e before they hit.) > > (5) When theory fails, resort to the real world. What happens to the > > kinetic energy of an electron and positron when they annihilate? > > Compare the energy of the emitted gammas to the initial rest energy. > > The difference is what the KE was. How does this compare with the > > result from (4). It's certainly much less than the result from (2). > > OK, we know that in the real world, the energy of the emitted gammas > is 8.19E-14 Joules. > > We know that the calculated kinetic energy of collsion should be > around 8.22E-14 Joules. No, we know that at this point, our classical calculations are wrong. Don't depend on calculations that we _know_ are wrong. Use measurements. Observed energy of annihilation gammas - rest energy of electron and positron = KE. Want higher energy photons? Just use higher speed (therefore higher KE) electrons and positrons. [cut] > So what is the answer to this mystery.... I'll give you a moment to > think about it... > > I don't know about you, but I would think the logical conclusion would > be that what we think is the energy released in the annihillation > event, is really just the release of the kinetic energy from the > collision of the positron and the electron. > > Before you go nahhhhh, can't be ... think about it. We just made a > legitimate calculation of the kinetic energy of a positron/electron > collision. No, we know it isn't legitimate. We know it's wrong. > It just happens to be by pure coincidence, the same as the > E=mc^2 formula eventhough the format and numbers of the PE formula and > E=mc^2 forumulas look nothing alike and you'd think there would be no > relationship. I just made that calculation while writing this post, > and there can be no better 'AH-HA!!' moment than that to see both > numbers line up. No pure coincidence at all, the classical radius is chosen to achieve this. (See http://en.wikipedia.org/wiki/Classical_electron_radius for some details.) Don't get too carried away by this "coincidence". Remember (a) measurements work, so use them to find the KE before collision, and (b) the electron is not a billiard ball with a radius equal to the classical electron radius.
From: PD on 12 May 2010 12:04
On May 9, 11:38 pm, franklinhu <frankli...(a)yahoo.com> wrote: > Let's say we have a positron and electron at rest with respect to each > other. Since they are oppositely charged, they are immediately > attracted to each other and begin to accelerate toward each other. At > any point in the path, you can calculate the force between them using > Columbs law, based on that, you could calculate the acceleration, but > how do you calculate what would be the velocity of the positron and > electron as they approach each other and then collide from some > starting distance like 5cm? As distance approaches zero, the force > approaches infinity. Does this mean that the velocity may approach > infinity due to the infinite force acting on a mass or is there some > limiting mechanism? Please help? > > -thanks > fhuemc Why don't you use potential energy converting to kinetic energy instead, Franklin? This is something that high school students learn to do. But secondly, I want to warn you that you cannot treat this problem classically all the way down to zero distance. This is not because the speed gets too high but because you are in the quantum realm, and so quantum mechanics has to describe the bound state. This is something a sophomore college student learns to do. PD |