PHGN-361 Spring
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Instructor information
Professor: Dr. Frank Kowalski
Office: Meyer 438
Office Hours:
Monday 12-2 and Tuesday 12-3.
Extra Credit Group Exam Problem Due May 2 at 10 AM
Please go to this site [1] (electric field of dreams) and play with the applet. Edit the following subsections iterating questions with answers. Generate both critical and positive comments on the questions and answers and come up with experiments to answer questions. All those participating will receive 20 extra points on exam 4 (there are 10 points per problem and about 5 problems on the exam) if the group draws insightful conclusions. Less credit will be given for less productive effort. You must log in on this wiki to participate. Please make your name apparent in the log in to receive credit.
Formal Q&A
Let's use this to make a formal question and answer section that follows "one question, one answer" format, and use the sections below as a workspace to iron out the details.
Q Does this follow the properties discussed during the 2-D infinite potential well in our quantum class? Also, how do different masses effect the observed field lines?
A Well, if we have just one particle, then our particle doesn't move around at all, unless there is an external E-field. This doesn't bode well with the 2-D infinite square well. The position and the momentum are known, and the particles aren't probability density functions. Then again, if we do put in an E-field, the particles are more likely to appear on one side than on the other because of their charge. This does change the expected value of the particles along the E-field lines. If a constant potential energy is added to the infinite square well problem, The particle does not (I think) behave any differently. It just goes about it's normal cycles as if the potential energy was at zero, which it is relatively. I do not think that these two problems are related at all.
Also, it does not seem as though the mass should change the observed field lines. The mass should only affect how the particles respond to the field lines.
Q: When many charged particles are inserted into the container, what is the temperature and what effect does the applied electric field have on the temperature?
A: The temperature is related to the average kinetic energy of the particles in the container, so to find the temperature of the container you would need to figure out the kinetic energy of the particles. To do this, you could to use the relations Work=Energy, Work(assembly)=Integral of V dq, and V=kq/r (where V is the potential and q is the charge). By increasing the electric field, the electric potential would increase, which in turn would increase the work done and raise the temperature of the system.
Q: When many charged particles are inserted into the container with an applied electric field what is the probability of finding a particle at a given position along the direction of the field?
A: Since the walls prevent the particles from exiting, there will be a high probability density along the wall opposite the direction of the electric field. My guess is that the probability then decreases exponentially in the direction of the electric field.
Assuming that the charged particles are electrons, we can begin with each particle being in a state (allowed excited or ground state) with a corresponding quantized amount of initial energy (dependent on the primary quantum number, etc.). The applied electric field will act on each electron. Depending on how much you would like to shake the electrons up and have them interact with each other (which sounds most unpleasant to calculate), we can also have a non-uniform or time-dependent electric field acting as an operator. Once taking care of the electron interactions with themselves, we can operate on each electron state with an operator of the form exp(-i(wt/2)Efield.PauliSpinMatrix). (Good guess on the exponential form, I'll guess that as well.) Then we can take that resulting state and take an inner product with our "given position" normalized state along the electric field and multiply by the complex square. That should give us the individual probability of one electron. But that sounds incredibly nasty to do for every single particle. We could also turn to the Fermi distribution and density of states functions for electrons with energy being pumped into our box from the applied electric field. This would be much more practical considering there are so many particles.
Q: Is energy conserved in this applet?
A: Energy seems to be conserved for a static electric field and no changes in number of particles. Changing the electric field can introduce energy to the system. Adding or subtracting particle also changes the size of the system and therefore the energy. Collisions with the walls of the box have a damping effect, and will eventually dissipate the energy in the system for a long enough simulation time. This was tested using a single charge bouncing off of a wall in a constant electric field.
Q: Does this model incorporate quantum mechanical effects and, if so, does the behavior of charges in the applet violate the Pauli Exclusion Principle?
A: The applet explicitly displays the position and momentum of the particles, therefore this is a purely classical view of point charge interactions. The effects of quantum mechanics and the stipulations of the Pauli Exclusion Principles are not applicable. Fermions are not prevented from occupying the same position, as seen when a neutral charge is introduced to a system of charged particles.
Q: Do 2 equal but opposite charges in a small electric field act how you would expect? Or does it seem that the size of the particles are drastically exaggerated relative to their motion?
A: Without the presence of an external electric field, the charges at first begin to behave how we would expect. They slowly attract each other until they hit each other and bounce off. After this initial collision it is difficult to tell if they exhibit the ideal behavior since they are moving so quickly. If we apply a small electric field, the particles will eventually exhibit the expected behavior. The positives will all eventually migrate to one side, and the negatives to the other. This effect is more easily seen if we apply a large electric field.
Q: Does this applet account for the magnetic fields generated by the moving charges?
A:No,this applet is only concerned with the Coulombic force between the charges and the force from the E field on the charges. As the charges speed up they do not appear to experience any additional forces, therefore we can assume that the magnetic fields are unaccounted for.
Questions
1) Energy Conservation: This reminds me of the fourth numerical assignment in Intermediate Mechanics. Do you think the total energy is conserved?
2) Chaos Theory: It appears that the location of where the particles are initially placed are not the same. So then does this system exhibit chaos? or is the boundary/walls of this system enough to predict the eventual outcome?
3) Electrons and Protons are fermions, but are neutrons (I'd hope so)? If yes then doesn't the applet violate the Pauli exclusion principle by allowing a neutron(assuming no charge in the sim is a neutron) to occupy the same space as the protons and electrons in the simulation?
4) Does this applet account for the fact that a changing electric field produces a magnetic field?
5) If we place tens of positive and tens of negative particles in this applet, could it act as a reasonable approximation for a confined plasma?
6) In the vein of 5), could we simulate the polarization of a material using positive and negative charges? Could a lattice structure of any stability be set up?
7) This applet made me rethink the apparent simplicity of Ohm's Law (J proportional to E). Ohm's Law tells us the current density given the applied electric field, but each charge in this applet visibly distorts the total electric field. It seems remarkable to me that the interactions of many, many charges could result in something that actually doesn't depend on these interactions. Just the applied field, as in Ohm's Law. So, a question I might have is: Does the collective motion of charges in the applet obey Ohm's Law?
8) With respect to the temperature increase by adding another particle: This would happen to increase the total energy since there is more potential energy in the system (because of the charge). However temperature comes from an increase in the kenetic energy of partricles in the system, so would this actually increase the temperature (as i think it would) or not?
9) How would the inclusion of the relativistic effects change the outcome the applet? How would the E-field change if we could move the particles near the speed of light? What effects could be seen and how would we account for this in the applet?
10) I've watched this video ten times now and cannot wrap my head around the implications that electrodynamics place on how the cats climb into the bowls in the ordered manner that they do. How does this relate to enm? Let's assume that the cats possess charge and spin. http://www.wimp.com/kittenbowls/
11) Again with energy conservation. I feel like the energy within this given system is not conserved because the walls of the box are always rigid. They give an opposing force without ever experiencing any changes. For energy to be conserved an external force of some kind would have to be acting on the box or at least there should be thermal energy being released from the collisions. However, assuming perfectly elastic collisions, energy would be conserved and no laws violated.
12)When there are many particles in the box it seems that the most probable place for the objects to be is on the outer edges. Is this just because of the repulsive forces of the particles or is there more at work?
13) Assuming the simulation is ran numerically, how does the sim deal with the situation that the charges are very close together and thus the the potential energy goes to infinity. Going back to the numerical assignment in mechanics, I recall that I just set a maximum potential that the objects can have. Is this how the sim deals with the situation, if so is there a better way to deal with it?
14) How would this Applet be affected if spin was introduced? How would the Pauli Exclusion Principle pay a role in this?
15) I am really curious as to how this applet would change if electrons were the charges. That would change the charge and you would have to treat the particles as waves. How could you represent the charge of waves and how could you show them change with time. Would you represent it with a complicated potential in schroedinger's equation?
16) Clearly energy is dissipated upon contact with the walls. Let's assume that the walls store all of this energy. How would you go about calculating the stored energy in the walls as a function of time?
Answers
1a) Energy conservation: I let the applet run for a few minutes, and no, energy is definitely not conserved. With three charges, it didn't take long before they were flying around the box with an obviously higher energy than they started with. The time resolution for the simulation must not be good enough...
1b) I believe Energy is conserved, setting the charge higher than the mass you can observe coulomb repulsion. Also I have not been able to observe the runaway effect stated above. I am curious however about how the collisions with the walls of the system effects this.
1c) After the applet had run continuously without intervention for about fifteen minutes, I couldn't detect any difference in the speed of the balls. As the balls interact with each other, their individual speeds change, sometimes resulting in one of them moving surprisingly quickly. However, it doesn't appear that the overall energy of the system changes over time and energy does appear to be conserved.
1d) I did two tests of energy conservation, with two of the simplest configurations. I first put a single charge in the applet with an applied E-field. After 5 - 10 minutes, the charge's motion had diminished noticeably, indicating energy loss. Same result for two charges. Conclusion: The charge-to-wall interactions are not conservative; they dissipate energy.
1e) If you set up a two particle system with one super massive particle (about 10^7 simulated mass units effectively fixing the particle) and one normal one, you can easily set up orbital motion in which the particle doesn't noticeably gain or lose energy. If you place the free particle very close to the fixed particle it oscillates very rapidly but neither flies off nor decreases amplitude of oscillations over time.
1f) In response to the statement above if you create a hydrogen atom type system with one particle having a factor of 10^4 more mass and starting at rest while the less massive electron is attracted to it. Than you can observe that the applet conserves energy as long as the particles don't overlap, but if they do the applet's calculations become inaccurate and the "proton" of the system will gain a large amount of kinetic energy out of nothing, of course as the mass of the proton increases the increase in kinetic energy becomes less pronounced.
2a) Recalling some information on Chaos from IM, the system can be completely "random" or settle into chaos. It is also difficult to distinguish between chaos and random noise contributions through numerical errors. Further analysis of this system is needed to determine if it shows chaotic behaviour or just random bouncing ball behaviour. Although at first glance, one could say that this system is chaotic. There is no long term pattern and the system is not predictable, thus satisfying the condition of chaos.
2b) In order for a system to be chaotic, it must, among other things, satisfy the following two conditions. 1. It must be deterministic, no randomness. 2. The predictions of the model are extremely sensitive to initial conditions. Assuming that this applet has no numerical errors or roundoff, which is likely a bad assumption, we can say that this system meets the first criterion. In order to test the second criterion I let the applet run for 10 seconds with nearly identical initial conditions. After a few trials, it was clear that a small change in location of one of the particles vastly changed the outcome of the simulation. Based on this information, I think it is safe to say that this system exhibits chaotic behavior
2c) Looking closely at the particles, it appears that 3 particles are all that are needed to create chaos. By restarting the system multiple times and tracing the paths of the particles when one is moved a small amount at the start. This shows that the system exhibits chaos, which aligns with the common perception of the 3 body problem. However, this assumes that there are no errors due to the truncation of numbers in calculations performed by the computer, which is a very flawed assumption.
3) This applet doesn't seem to apply anything related to quantum mechanics. It wouldn't make sense to simulate electrons, protons, and neutrons as point particles if we were interested in things relating to quantum mechanics; we would need to use wave functions. With that said, the Pauli Exclusion Principle states that no two fermions can exist in the same quantum state i.e. have the same quantum numbers. This doesn’t mean that there is no probability that the two can’t be in the same position. So even if the applet used wave functions and showed a probability that the two had the same position, it wouldn’t be violating the Pauli Exclusion Principle as long as the two still had different quantum numbers.
3b) In addition to the statement above even if we assume that the applet displays the expected value of position and momentum (ignoring Heisenberg's uncertainty principle) the Pauli Exclusion Principle only applies to wave functions so even if they did have the same expected position and momentum then as long as there is a factor that makes their wave functions different (such as spin) then there is no violation of the Pauli Exclusion Principle.
3c) The Pauli exclusion principle cannot really be applied to this simulation as it is presented to us. In quantum we learned that the four quantum numbers determine the state of a particle and we are missing one of those descriptors. The pauli exculsion principle arises from the anti-symmetry of the wave functions of fermions, wavefunctions composed of psi and chi, and given we have no input from chi, this means that instead of the simulation violating or upholding the exclusion principle, it simply ignores it.
4) The Electric Field of Dreams does not account for magnetic fields produced by a changing electric field. This is evident in the description of the applet and the behavior of charges within the applet. The explicit description of the applet states that the learning goals are to explain "the relation between the size and direction of the blue electric field lines to the sign and magnitude of the charge of a particle," "the interactions between two charged particles and explain why they move as they do," and "what happens when you apply different external electric fields." Within the applet we find that the only term accounted for in the Lorentz force law is the electric term. There must not be any magnetic force because there is no behavior exhibited by the particles indicative of the term containing the cross product between the velocity of the particles and a magnetic field. There is a discontinuity any time a charge is added.
4b) I agree that the applet doesn't exhibit any visible relation to the magnetic field produced by a changing electric field. If it did, we would likely see the particles behave differently as they move faster, since the force on an particle is proportional to both the velocity and the magnitude of the magnetic field. For example, if you put two particles of charge +/- one but of large masses (~10) and watch them interact, they accelerate at a fairly uniform rate. It would seem to me that if the applet accounted for the magnetic field, as soon as one of the charges started moving it would generate a magnetic field, which would instantly put a greater force on the second particle, causing it to speed up much faster. Since I didn't see this effect, I am going to assume that the applet does not take magnetic fields into account.
4c) Call particle (1) the 'free' particle and particle (2) the particle that will create the B field as we move it. By dragging (2) in a straight line in the +yhat past (1) which is to the right we create a system analogous to an infinite wire since (2) just became our I. From this we know by the RHR that B will point into the page and we know(directionally) v since we are dragging it. Solving for F we get q[E(in +xhat) + v X B (resulting in -xhat)]. The B field created therefore opposes the E field and what we should observe is that the dragging of (2) past (1) should actually slow the repulsion if not create attraction(based on the magnitude of v). This does not happen on the applet and if you move (2) fast enough the script does not even show (1) reacting to the presence of (2).
4d) It is important to remember that the magnetic field is much smaller than the Coulomb force. It is hard to detect unless the Coulomb force can be negated, as happens with current flowing through a wire. That said, I looked at the source code (phet.unfuddle.com, java-simulations ->efield) and there is no magnetic field.
4e) In this course we never dealt with magnetic fields produced by individual moving charges. We dealt with currents. The reason that magnetic fields from currents that move at non-relitivistic speeds are of non-negligable magnitudes is because of the huge quantity of charges involoved in a current through a wire. Even if magnetic fields were accounted for they would not have a noticeable effect on the behavior.
5) I think to provide an approximation for a plasma would a far more complex setup than the applet can provide. Even as high as 20-30 different "atoms" in the simulation start producing some weird results (particles jumping from one side of the box to the other almost instantaneously). I'm currently trying to get a system to "settle" into an equilibrium state with ~50 of +/-1 charges of mass 1, and nothing seems to be changing. I think we would need on the order of hundreds, if not thousands, of particles to successfully approximate a plasma, and certainly in a larger container. The "boundary" conditions of not being able to go outside the box seem to be messing with any good approximation that we could get from this applet.
5b) As the applet seems to fail if ran for a time with even a hydrogen atom like system it seems unlikely that applet would become more accurate if you added more particles to the system.
5c) From the definition of a plasma (http://en.wikipedia.org/wiki/Plasma_(physics)), the applet wouldn't provide a good simulation. Although there are multiple particles interacting with each other (the plasma approximation requirement) it is debatable whether the applet could provide a short screening length (bulk interaction requirement) given the large size of the particles compared to the simulation area. Also, the particles used in the simulation are much larger than I would expect from physical particles compared to the fields created. This means that there will be many collisions between the particles and, as already noted, the simulation handles collisions poorly. Also, many collisions violates the third property of a plasma (the Plasma frequency requirement).
6) A self-stable lattice would be very difficult to create using this applet. There seems to be a chance of a rounding error of sorts when a negative charge gets too close to a positive charge and the force between them blows up. There also seems to occasions when energy is added to the system even with no external electric field. It's also very difficult to tell if a lattice is stable or not in this applet. It would need to be composed of many atoms, which leads to a cluttered viewing screen. In order to create a stable nucleus, we would need strong force. In order to create a stable latice, we might need a larger and more precise simulation.
6.b) If I understand this correctly, this is similar the question of trapping a particle in a box is it not? We know this cannot be done since we can't create a restoring force as a result of grad dot E = 0 (since we are looking for an 'electrostaticle' condition).
7) I was thinking about the question when I was looking at the applet as well, so I decided to try and approximate the surface of a conductor by placing a bunch of -1 charges of equal mass in the box. They bounced around for awhile, then eventually settled on the outer edges of the box. One of the limitations of the applet is how it behaves with a lot of charges, but additionally I found that this applet does not seem suited to answer this question. When we are looking at how J relates to the applied E field, we are talking about a current density of sorts. While a current is essentially moving charges, the applet does not have any sort of extended geometry to allow for these charges to go anywhere except in the confines of the box. Therefore, we cannot see how the true current density behaves because we cannot approximate any sort of "current" (at least I couldn't). No matter how many particles go in the box, it still acts like a system of point particles rather than a conductor or other more complex material.
8) The statistical definition of temperature is the mean kinetic energy of the particles. The mean kinetic energy doesn't depend on the quantity of particles in the box. I assume that each new particle added to the box has the same initial kinetic energy (I guess I could check the source code to confirm that), so the number of particles added to the box shouldn't affect the temperature.
8b) If you look at adding in an Electric field to the system with a number of particles in there, it seems like although this would increase the voltage and in turn the total energy of the system the charges get pushed to the sides of the system where they seem to loose a lot of their kinetic energy. If this is really the case then adding a stronger field would slow the velocity of the particles and therefore decrease the temperature even though it adds energy to the system.
9) The inclusion of relativistic effects would create electric field lines similar to the "Field of a Moving Point Charge" applet that we played with earlier. The field lines become more dense perpendicular to the motion of the charged particle due to length contraction in the direction of motion of the charge. To account for this in the applet, the velocity (speed and direction) of the charge would need to be taken into account, so that the length contraction can be calculated and the field lines can be altered as necessary.
10) Interesting question. I'm trying to wrap my head around it. I think its fair to treat the system quantum mechanically as you don't see two cats in one bowl and thus pauli exclusion is satisfied. Extrapolating from this assumption I conclude the only way we could get two cats in one bowl is if they were laying head to toe (assuming they're fermi-cats that is).
11) The collisions seem to be perfectly elastic (or at least they are supposed to be). This is just a simulation, and is definitely not a perfect one. We can just treat the walls as being very massive so that they don't have to move when the particles hit the wall.
11b) Earlier questions talked about the increased temperature of the system due to the impacts of the particles with the sides of the boxes, which is a reasonable, physical explanation. It does however, require that the collisions with the wall transfer kinetic energy into thermal energy meaning that the collisions cannot be elastic, even if the wall is treated as massive. In this case energy may be conserved if the walls are included in the system, otherwise the particles should lose energy on impact, and energy is not conserved if the particles alone compose the system. The treatment of simulation does seem to follow perfectly elastic collisions though, since I could detect no changes in kinetic energy after impacts with the wall.
12) When the particles have they same charge, they are naturally repelled. However, the walls prevent them from getting too far away, and causes some problems like bouncing and such, but this is the general idea to why the particles are mostly found of the outside of the box.
12b) They are naturally repelled, and the box does keep them inside, but the charges in the middle are still the farthest away from the other charges as they can be. I feel like the configuration in which most of the charges are near the outside is the most favorable position they can be in.
12c) It is probably incorrect, but when I thought about this, by analogy, I considered how charges spread out on the surface of a conductor. The charges will repel, and seek to be as far apart as possible which is a distribution with higher probabilities of particles on or about the surface, in this case the interior of the box, with the particles being more likely in the corners. The external electric field's magnitude and direction will influence this distribution, but in a similar manner to how the distribution is changed when an electric field is applied to a conductor.
13)a) I believe that they would define radius of the particles and then have a defined value for when the particles are within that value. This would show a degeneracy effect
13)b) In classical mechanics we used point particles and the infinite potentials arose from the particles approaching unrealistic distances. We weren't dealing with singularities when we created that model for the 3-body problem, so the model broke down for those close distances. As for charges, it seems reasonable for them to get incredibly close if they are small in size (electrons). The problem is resolved if you instead think of their wave functions and not their absolute position as point particles. Just look up solutions to the infinite square well with multiple interacting electrons. I believe these solutions are used for some chemical molecules (we saw some of this in quantum on wednesday). As for the app, they most likely keep the force constant at some distance so that the charges don't fly off at instantaneous speeds.
14)a)If spin were involved, then the electrons would still be repelled by each other, but they wouldn't be able to be in the same state. I guess I don't really know what "occupying the same state" really means, but from what I assume it means, they cannot be in the same position. In this applet, they seem to be able to be in the same position every now and again. At which point the repulsion becomes very large and they just fling apart. If they were actually electrons, I think they would just "blink" out of existence if they occupied the same state. This would be a much funner applet if this were the case!!
14)b) If spin was introduced I suppose you would just get some electron degenerate pressure that would fight against the e-field changing the way the electrons would organize themselves. If the force from the e-fields overcame this pressure the electrons would decompose and lose charge which means the particles would no longer impose an electric force on each other (maybe one electron would decompose to relieve some of the degeneracy pressure and electric force)? Also, as this force increases the electrons get more and more speed (think compact space and the uncertainty principal) which means the electric force would have to be pretty massive (tightly compacted electrons). On top of that the potential of the 'wall' constraining the electrons to this box would have to be infinite to contain these huge energies.
15) You can choose the charge in the applet by clicking "properties", and one of the options is -1 (an electron). A probability density function could be displayed in the applet. It could be a bunch of shaded regions, where each color or line represents an equipotential surface. For example, for one electron, the expected value of position would probably be somewhere near the middle of the box, depending on the external field. So the applet could display a dark circle in the center of the box with concentric circles around it, getting lighter and lighter as they approach the walls. (They would probably not be perfect circles, due to the applied E-field.)
16) It may seem a little odd, but the means I would take to find the energy lost by the collisions would be through thermodynamic principles. Pressure is the measure of collisions of particles in a system, which is what you're looking to find the energy of. From this, the energy lost can be found since pressure multiplied by volume is energy.
Critical comments on the answers
1a and 1b) It's worth noting when talking about energy conservation that we can apply an external electric field to the box. An electric field can add energy to the system. Thinking about the scenario where there is just one charge in the box at rest, when you add an external field the e-field does work on the charge and moves it. This could be how the system gains energy with time.
1c) The fact that there was no visible change in energy in a system with multiple balls does not necessarily mean that the applet obeys conservation of energy. It is difficult to detect a change in energy in a system with multiple particles as all forms of energy for each particle have to be determined.
1c2) When two particles are placed in the simulation with equal mass and equal but opposite charge and the default electric field, the particles begin with relatively low kinetic energy and take a couple minutes before they are able to get around each other and interact, due to the fact that the electric field makes them act in opposite directions. But once these particles are able to interact with each other their kinetic energy is greatly increased, due to the Coulombic interactions. If left for even a couple more minutes, the two particles begin to move in a uniform circular direction, hitting two walls per revolution. This behavior is not chaotic in motion and very predictable over time. The circular motion then evolves into the two particles bouncing in two straight lines from right to left, one particle above the other.
1e) The inclusion of a "super massive" particle shows the fact that the applet does not account for the gravitational interactions between two particles. As the mass of a particle becomes sufficiently large in any system the gravitational force quickly overrides any Coulombic interactions between the two particles. When the mass of the interacting particles is sufficiently small, the gravitational force between them is many orders of magnitude smaller than the Coulombic interactions. It would be interesting to see where the gravitational forces overcome the Coulombic forces if they were included in the applet.
Q:
2) I found that it was easy to deliberately add or subtract energy to or from the system by altering the direction of the 'external' electric field, but without making any alterations to the field, it didn't appear that the energy of the system was changing with time.
4) If this were an actual simulation, we would, unfortunately, not have an applet description that could tell us which physical phenomena to take into account. Also, how do we know the electric term of the Lorentz force law is the only one accounted for in the applet? What behavioral change would we see? Would a change like this be large enough in magnitude to be noticeable?
6b)Trapping a particle was possible in 2 dimensions which we found in homework 5 (in 2D the grad dot E was not equal to zero and the potential had a local minimum). That is, Earnshaw's Theorem applies in 3D, not in 2D. By setting up four massive particles and one free particle it is fairly easy to trap a particle.
8) It seems to be agreed upon that particles lose energy when they make contact with the walls. Therefore, if we define the system to be the particles (but not the walls), the average kinetic energy of the system is decreasing over time if the particles are moving. Assuming each particle starts out with the same kinetic energy, T(not), which decreases each time it hits a wall, then T(not) > T(t>0), which means T(not) > T(average). Therefore, each time a particle is added, the temperature of the system increases.
13) Assuming that the force between the two interacting particles is constant when the particles are in close proximity to one another is a valid estimate. This approach doesn't account for the change in direction of the particles - it would allow the two particles to pass through one another. This could be corrected by setting a minimum allowable separation distance between any two particles and assuming all collisions are perfectly elastic (still not the best approximation to the motion, but a step forward). By accounting for the collisions, the speed and direction of each particle can be maintained (eliminating the infinite force given at 1/r when r=0).
14a) Wouldn't you then have to take into account the wave functions. Then you would have all possible states exhibiting themselves and their energy. You have have to take this effect into account. So everything that the electron could do would have some representation. Unless the wave equation is broken down. Then it seems that what you said would be true
Positive comments on the answers
1) 1b seems to mirror the situation I encountered with the applet as opposed to 1a, that is if energy conservation is defined as initial energy equals final energy. As long as the electric field is constant (such that no external force acts upon the system) then the energy should be conserved.
2) From looking at my app for a good ten minutes now, I can say that it seems to be a chaotic system. I think the amount of balls may play into whether the system becomes chaotic or not (12 balls result in a chaotic system).
3) As far as creating a stable lattice system, I think that this applet may have some of the limitations were difficult to work out in the 4th Mechanics numerical as well- mainly that the Coulombic force blows up unless you program it correctly. If there are too many atoms with opposite charges in the simulation they just start jumping around and I think some of the physical meaning is lost.
7a) It is good to point out that in order to observe or test Ohm's law we need to see a current. Since there is not a well defined dQ/dt which is usually associated with a dl how could we look at the law? Additionally, I would ask what can we observe about R? Since the charges continue to move like pucks on an air hockey table (i.e no friction) does the applet have some implied resistivity? My guess would be no, the applet does not have a built in resistivity (using the air hockey table argument).
7b) I like the way the contributor above analyzed the Ohm's Law question. Their response made me realize how this applet is unsuited to answer questions on Ohm's Law. If you want to relate J to E, then you need to build a model of a whole circuit; you can't have the charges confined to a tiny space like this, otherwise you get charges building up everywhere and can't sort out the effects of the E-field from the effects of the bounding container. For me, I am still interested in how Ohm's Law originates in a given material. I mean, it starts with a retarding force on the charge that's linear with its velocity, but where does this linear relationship come from? Could we derive it classically with a billiard ball model and the Coulomb force? Or is it a QM result that has to do with the charge's wave function and energy bands in the material?
10) I like you're answer, but it begs the question: what about bosonic cats? distinguishable cats? These are situations that must be considered in detail. I think the calculations would be relatively straightforward, but we should try to come up with experiments to verify our predictions.
Experiments to test questions
1a) An easy experiment to test conservation of energy is to start the system with 2 particles and no external electric field so there is motion and then remove one. We now have one particle bouncing around all alone that should have no external forces acting on it. If we let it run for 15 minutes or so and return, it is traveling at a noticeably different speed. This would indicate energy is not conserved.
1b) I think it would be interesting to go back to our 3-body problem Mathematica notebook and generalize it to n bodies. If we set the timestep small enough, allowed random starting positions, and kept the "if r<r0, r=r0" statement, we could see how well it compares to what this applet is doing. I suspect we will find our notebook does a much better job approximating either way. If we get the timestep small enough such that large velocities will not move particles significant distances within the timestep, we could even remove the r0 statement, allowing us greater insight into the actual situation.
1c) A definitive way to show there is roundoff error, or some other factor, effecting the field energy is to set up a cage of charges around the edges (use the pause button). Once this cage is created, 16 charges in this case, a single charge can be put in the middle and given some small initial velocity. Instead of oscillating indefinitely the amplitude dies off. Notes: The "cage must be stable to set up the experiment. The results are much better using a pc instead of a laptop.
1d) An experiment that I performed was to take two charges of mass unit 1 and opposite charge. The two masses were placed near each other so that they would be held together by the Coulomb Force. After leaving the applet for approximately 30 minutes, the charges had separated and were bouncing around inside the box. For the two charges to have separated, work needed to be done, and the Work Energy Theorem then tells us that this work changes the total energy, thus telling us that the total energy is not conserved.
2a) To see if the system exhibit chaotic behaviour, we can run calculations and analysis through Mathematica (as in IM Numerical Assignment 5?). Also we can run multiple simulations simultaneously and watch to see if the system runs in the same path at some point.
2b) A specific experiment to test if this simulation can exhibit chaotic motion would be to set up two nearly identical situations differing only by a tiny preturbation in the initial conditions and start them at the same time. If they follow very similar patterns of motion then the system is not chaotic. If, however, the slight difference in initial conditions causes a huge difference in motion later on then the system is chaotic. Exe: Two situations were set up with charges of mass 1.0 and charge -1.0 in the top left and bottom right corners with no external electric field. The initial positions of the charges varied by only a few pixels from one set up to the other. These simulations were started at the same time. Around 5 minutes into the simulations, the charges in one simulation were clearly moving in very different manners than the charges in the other simulation. This would imply that a system with this set up does indeed exhibit chaotic motion. However, by an hour into the experiment, both applets exhibited the same pattern of motion: the charges move back and forth horizontally with very little vertical motion. It would be interesting to test if this final pattern of motion holds true for all two charge systems.
4) See 'answer' 4.c) to test if the applet accounts for B
5a) If you set the mass ratio between a positive charge to a negative charge to be the same as the ratio between actual protons and electrons (about 10^3) then you can set up a hydrogen-like atom. If you get a bunch of these with all particles randomly spawned, the negative charges will tend to orbit the positive charges as the positive charges just bounce around. However, once the number of negative charges exceeds the number of positive charges, the negative charge will tend to float around until it bumps into a hydrogen "atom" at which point there is a chance of one of three thing happening: both particle leave the "proton"; the "electron" is just deflected; the incident "electron" knocks out the other electron and new electron remains in orbit around the proton. However, you can "ionize" the gas by just giving the atoms a bunch of initial velocity. The larger the average kinetic energy of the gas the larger chance it has of ionizing the particles. So Yes, this does seem to act like a confined plasma as the temperature increases.
6a) One can set up a lattice using massive particles (on the order of 10^7 simulated mass units). This doesn't accurately represent a real lattice since protons and electron vary in mass by only about 3 order of magnitude but using super massive particles allows for the following experiment. I made a lattice of massive particles with layers of alternating charge to represent a polarized material. I then let the simulation run to watch what would happen as free negatively charged particles spawned in random places. Any particles that spawned on the positively charged side of the lattice remained trapped on the that side. All other particles were pulled into the lattice and eventually paired up with a positively charged massive particle (like an electron orbiting a nucleus. However, once all the positive charges are paired any new particles will be repelled to either side. I noticed something interesting once the lattice is fully paired; by giving new particles some initial velocity they will pass through the lattice as though it weren't there unless it gets near another negative charge already orbiting a positive charge in which case it will knock the other negative charge out of the lattice and "ionizing" the positive charge. This shows an interesting effect that accurately relates to macroscopic objects in the real world. By increasing the average kinetic energy of the system there becomes a larger chance that negative particles (like electrons) will be freed from the lattice. You could relate some sort of ionization energies for the simulated atoms to the "temperature" of the system. So to answer the question, yes a lattice can be set up and it does interesting stuff, but it requires unrealistic mass ratios between the "protons" and "electrons".
6b) Regarding the contributor who wrote the above comment "6a": Friggin' sweet. I tried running your setup with a small "chunk" of material made of +1/-1 charges, and it works like a champ (incidentally, I ran it using positive particles with a mass like 10^9, but that's even more unrealistic). But seriously, we can start classically modeling the behavior of all sorts of materials now, we can even go through the whole periodic table from Lithium (+3 nucleus), Boron (+4 nucleus), etc. We could even try simple molecules like CO2 and see what the simulation gives us. Throw in a lightly-charged particle (neutral particles don't interact very often, if at all, in this applet) and model the photoelectric effect. Build two lines of charge and see what the fringing fields around a capacitor look like. Seriously, dude (or dudette), who wrote 6a--bravo. Nobel Prize worthy.
8) Since it would be extremely difficult to measure the velocity of each particle, we could roughly define the kinetic energy (which is proportional to temperature) as being proportional to the rate at which particles hit the walls divided by the total number of particles. Now put a bunch of particles in the box and then record the result (by video perhaps), occasionally adding particles. Watch the recording in slow motion and count the number of interactions with the walls (or rewrite the applet program to count this). See if it increases when you add more particles.
9) To test if the applet did take into account the relativistic effects, I took a particle of high charge and very low mass. I chose a large value for the charge so that the field lines might be easier to see and low mass so that it would be easier to move. I sent the charge moving first from left to right and then from top to bottom, decided that didn't really help me see anything, and then decided to send the charge diagonally. With the charge moving very fast, and the field discreteness very high, the field lines did not look distorted. The field lines looked as though someone had taken a snapshot of a stationary charge, moved the charge into another position, taken another snapshot of it not moving, and continued this over and over again, and then played it like a flipbook. From this I am going to conclude that the relativistic effects are not included in this applet.
PROB of POSITION) I ran a test to try to find which factors affect the probability of finding the position of a particle as asked in the Q&A section. I first ran the applet with 24 charges. The charges isolated themselves, with most scattered around the edges of the box and oscillating in only one dimension (or very little in any other), and around 4 in the middle oscillating in circles but still confined to a small region of space which the other charges did not enter. The charges were essentially trapped. I did the same test again, but with 11 charges. The same effect was observed, which was expected, but it took time for the charges to find a space. This leads me to believe that the probability of finding a particle is affected not only by the size of the box, but the number of charges (which was expected), and the time that the applet was allowed to run.
Comment related to the above test: this would make it appear that the confined regions are points of minimal potential energy, and a stable equalibrium. Which would lead me to conclude that the charges allign allong the wall with the e field until their combined repulsive force counteracts that of the e field, causing additional charges to hover around in circles, something to test would be how the amount of charges total effects how many on the wall versus in the center, aka 12 charges =8-4, does 10 charges =7-3?
13) You could test the hypothesis that they set a minimum radius for how close the particles could interact by setting up a series of experiments within the applet. If you start the applet with the charges a specified difference away and then incrementally decreased that distance, you might be able to find a range at which the particles don't act in accordance with Coulomb's Law. I think this would actually be pretty difficult though because the applet does not have specific distance markers and the specified distance is probably much smaller than what we could control simply using the mouse to drag the particles around.
Other comments
1) Energy Conservation: Something just occurred to me, which I didn't think of when I was working on the fourth numerical. Total energy could be maintained (even though there were obvious spikes in the total energy when the three masses approached one another) by normalizing the energy at each timestep, just like we do in quantum and thermal. That is to say, reduce each objects kinetic and potential energy by a factor such that their relative energies remain the same, and the total energy is constant. The only issue with this is that divvying up the energy into kinetic and potential for each particle might be a little tricky (especially because the potential depends on where each particle actually is. This would be a cool feature to go back and add to that three-body simulation, though.
2a) The details of this simulation is limited, ie. we don't know the accuracy of each calculation made and we don't know how many decimal places the calculations keep track of. All we know is that there is repulsion between the two balls, we can set the amount of repulsion/mass through the preferences button, and the total amount of particles in the box.
2b) In regards to the chaos idea, if someone else is finding that it eventually settles into the same motion for two particles, we might be able to make the assumption that after a long time, all two-particle systems would exhibit similar behavior, meaning it would not exhibit chaos. Remember, chaos is when two systems diverge and never come back, whereas we can have systems that exhibit seemingly random and divergent behavior, but come back to the same set of parameters after a long time. In the IM lectures on chaos, remember that we truncated many of the initial cycles.
3) I found it interesting that when I added a particle with no electrical charge, the other (charged) particles didn't interact with it at all. It's clear that the charged particles interact with each other only due to their charge (since they don't actually impact each other), but I thought that the program might allow collisions between particles in the absence of electrostatic repulsion.
4a) Along the same lines as collisions, I am pretty sure that this applet does not account for magnetic fields as well. Assuming only two dimensional motion, the "currents" created by the moving charges would produce a B-Field that would either attract or repel the other charges. Just an interesting note.
4b) It is tough to gauge the presence of a magnetic field in an applet that works with two dimensions. Think about the cross products involved in finding the magnetic force using the Lorentz force law and the magnetic and electric fields using Faraday's law and Ampere's law.
5) Here's an interesting note: Crank up the field resolution (Electric Field Menu at the top of the applet) as high as it will go. Place a single charge. Click and drag it around the screen (even outside the box) while watching the E-field. Note that the field is not circular! What's up with that? Answer: Before you put a charge in note that there is a fixed grid where the field lines are calculated. Once you put the the charge in grid more blue appears in the corner regions because the field lines do not line up with the next point in the grid and so look more dense.
Other.a) After playing with it, it appears that the applet does not give particles any radius or physical presence. I was testing with collisions of massive, neutrally charged particles and they did not interact. I asked this when the particles were still charged and I was getting huge scattering velocities at very close radii: maybe a similar code to our Intermediate Mechanics with the multiple body system and how we had to modify the code to handle the singularity cases.
6) I have a strong hunch that this applet uses some sort of iterative method with a set time resolution. Since this program cannot use an infinite number of time steps when calculating the forces on any given particle, there is a limit on how accurate it can calculate the motion of the particle. For the conservation of energy experiments that many people did, this means that based on how you set it up, energy may or may not be conserved. A quick experiment is to apply an electric field on a single particle straight down, and release the particle somewhere in the middle. If the particle's charge to mass ration is too great, it will approach the wall of the box fairly quickly. If it is moving too fast, the program will miss the point where the particle hits the wall and bounce back, resulting in some sort of computational error in the particle's speed. My guess is that if the calculated position of the particle on any given iteration is outside the box it moves it back inside the box and applies a calculation for it elastically bouncing off that wall. This results in the particle "losing" some of its kinetic energy. This means that the particle will bounce back lower until it reaches a point where the incoming velocity is slow enough that the program can accurately describe its motion. This would explain the vibrating effect of particles on the wall when the charge to mass ratio is really high. On particle/particle interactions, since the particles are given no effective volume (I agree with "5 Other.a" on this), if the (similarly charged) particles approach each other at high enough speed, the program will miss the turning point where the two particles should scatter and will calculate a force on the particles in a location that is much closer than they would have normally come causing them to fly apart at ridiculous speeds.
7) When you apply the external field perpendicular to a side of the box, the equation of motion for a particle will look like a damped harmonic oscillator. (If you play with the charge and the strength of the field you can get it such that there's very little damping and approximate it as an undamped harmonic oscillator.) In that case we would get a frequency of oscillation that is based on the charge and mass of the particle, and the strength of the field (Given that the collision of the particle and the wall is elastic). So if there was a situation in which we wanted to experimentally determine one of these three parameters, we could do so from the motion of the particle.
8) After thinking about the total energy of a particle, it seems like this applet is very limited in its scope. It would be interesting to see how the addition of more energy terms (characteristic thermal energy, relativistic corrections, gravitational attraction, etc.) would affect the applet. Even if most of these can be neglected for the short time case, they would create a distinctly different outcome over long periods of time.
Formal Q and A about prob. of position) The strength of the electric field varies with 1/r^2, so the probability of finding a particle at a certain point would also be related in this manner. If we were to know a function of the position for the electron, we could find a probability density function by normalizing it (much like in quantum with the wave function).
9) I found it interesting what occurs after running the applet for an extended period of time with two particles of charge -1 and mass 1 in the box and no external electric field applied. The particles end up on opposite sides of the box constantly in anti-parallel motion. I wonder if the particles would settle into opposing corners of the box if allowed it were to run even longer or if this motion would continue. If the motion continued, would it be because of the initial conditions of the particles or would it be due to program coding reasons?
10) In 2D it seems I should be able to trap a positive charge in the center, with 4 negative charges at the corners. But the program doesn't allow the charges to remain at rest, and after a short while, the charge is no longer in the center. Is this because of physics of computer programming? It seems like it's probably because of the time steps in the iteration.
11) Regarding comment 9: This parallel motion is interesting - it seems that for the 2-particle case, we should be able to find an analytical solution for motion (like we can with gravity), and in such a solution we'd probably find periodic motion. But this applet seems to use only conservative forces, meaning that no energy is ever dissipated from the system, so this motion should continue indefinitely (which probably wouldn't happen in a real system, because those oscillating electrons would produce radiation! Regarding comment 10: This is also interesting. We talked about trapping charged particles using other charged particles earlier in class. A charged particle cannot be trapped by other charged particles, because we cannot create a potential well to trap the particle in! (Because Gradient(V)=0 in free space, the charged particle we want to trap will never find a real potential well.) So the fact that this applet preserves this physical reality is pretty impressive.
Creativity Links
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Course Information
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Course videos, applets, and links
Video Lectures for March 7 through March 11 are
Here is the InkSurvey link Kowalski InkSurvey site
Homework assignment 7 has for problem 1 this link applet
and for problem 2 this link cell phone
Homework assignment 10 uses the following link [2]
Applet links:
Harmonic oscillator applet [3]
E from a moving charge [4]
Charge and field distribution on conductors and in dielectrics [5]
metal sphere inside a capacitor where the voltage across the capacitor can be varied [6]
Applet link for Faraday's Law used in the April 13 lecture [7]
Applet link for Faraday's law with a rectangular wire near a long wire [8]
Applet to calculate inductance [9]
Plinko link [10]
Lectures
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Homework Assignments:
See the syllabus for the first homework assignment.
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Homework Solutions:
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HAARP problem in assignment 12.
A part of rhetoric [11] is to manipulate the facts to sway the viewers’ opinion. You need to recognize this manipulation in the context of science and engineering (not only in politics). One example in this video is that interruptions of communications in a nuclear war are the primary objective of HAARP.
Indeed nuclear explosions generating an electromagnetic pulse (EMP) do disrupt communications by creating gamma rays whose Compton electrons spiral around the Earth’s field lines thereby producing a huge EMP (large enough to fuse 300 street lights 1300 km away). Please see the following references [12] and [13]
However, HAARP is not a nuclear weapon. Yet that doesn’t stop the authors of the video from making that false connection. Here are more reliable descriptions of HAARP along with a discussion of conspiracy theories associated with it.
Other manipulations of the facts are that the Earth’s rotation rate will be dramatically affected. It is easy to calculate the energy required to do this and compare it with the electromagnetic energy that can be generated.
Just like in the applets you need to think critically about the information you find. Here are some resources
-the effects of HAARP [14]
-HAARP on wikipedia [15]
Exams with solutions and Rubrics
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Old Exams with solutions and Rubrics
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Problem solving strategies and sample problems:
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Volume and surface integrals in cylindrical coords [16]
Line integrals
Delta functions
The secrets of gardening tips [19] for totally tomatoes
