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: Is energy conserved in this applet?
A: Energy seems to be conserved for a static electric field. Changing the electric field can introduce energy to the system. 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: This is a purely classical view of point charge interactions. Therefore, 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.
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?
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.
2) 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.
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.
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.
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.
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.
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 taht 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.
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.
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?
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.
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.
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.
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?
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
