Phys 3 Lab 8 Ch 1 Interations with Magneti Fields 1 Equipment: omputer with VPython, single e/m apparatus for qualitative experimenting: Fore on dipole: blak power supply, TeahSpin Magneti Fore apparatus, high-watt 1 resistor, digital Multi-meter, BNC ables Objetives In this lab you will: Simulate the fairly-uniform field of Helmholtz oils Simulate the motion of harged partiles in a uniform field Qualitatively experiment with eletrons moving in the fairly-uniform field of Helmholtz oils Qualitatively experiment with dipoles interating with the field of Helmholtz oils Simulate the smoothly-varying field of anti-helmholtz oils Qualitatively and quantitatively experiment with dipoles interating with field of anti- Helmholtz oils. Interation with a Uniform Magneti Field I. Produing a Uniform Magneti Field Helmholtz Coils a. Theory Before experimenting with or simulating a harged partile moving in a uniform magneti field, we ll digress and see how suh a field is typially produed for suh experiments. As you re familiar, a solenoid produes a very uniform magneti field; reall the simulation and measurements you made in Lab 5. In pratie though, a omplete solenoid is seldom used for one thing, it s awkward peering inside to see what the harged partiles are doing. With the right geometry, you an still have a fairly uniform field even if you remove all the oils from the mid-setion of the solenoid, leaving idential sets of oils only at the two ends. This right geometry is having the distane between the end oils equal to the radius of the oils. Pass the same urrent through the two sets of oils and you have the Helmholtz onfiguration. I I B R The field strength at the enter point should be R I, (1) R o Bmiddle 5 3/ 4 This follows from the Magneti Field of a Loop in setion 18.8, multiplied by sine there are two, equidistant sets of oils, and with z = ½R. What would be the field strength if you had a 1 amp urrent and a radius of 0.3 m? Note: In pratie, eah of the two remaining urrent loops is itself usually a set of N losely paked oils of wire; so the urrent through one loop would be N times the urrent flowing through just one wire.
Phys 3 Lab 8 Ch 1 Interations with Magneti Fields b. Simulation So you an visualize the field produed by a set of Helmholtz oils, you ll modify your Solenoid simulation to produe a HelmholtzCoils simulation. Download the opy of the Solenoid.py simulation available through WebAssign. As usual, open a fresh instane of VPython, paste the ode into it, add your names in a omment line near the top, and save as Helmholtz.py. For the sake of omparing the fields of a solenoid and of Helmholtz oils, before modifying the program to simulate Helmholtz oils, add a few lines so the existing program plots the strength of the magneti field aross two axes: lengthwise along the solenoid s entral axis, and perpendiularly aross its middle. It s been the better part of a semester sine you ve done anything like this, but you used similar lines of ode in Phys 31 when plotting things like the momentum of an alpha partile sattering from a gold nuleus. To be able to make rudimentary graphs, at the beginning of the program add the line from visual.graph import* To prepare the program to make the graphs, in the objets setion of the ode, add the lines Bygraph = gurve(olor = olor.yan) Bxgraph = gurve(olor = olor.red) To add points to these urves, at the bottom of the for obslo in obsloations: loop (below but outside of the for y and for theta loops), add if obslo.x ==0: #plots y omponent of B for points along x=0 axis, along solenoid Bygraph.plot(pos = (obslo.y,b.y)) if obslo.y == 0: #plots y omponent of B for points along y = 0 axis Bxgraph.plot(pos = (obslo.x,b.y)) Run the program and observe the width (in x, red plot) and length (in y, yan plot) of the entral region in whih the field remains fairly onstant. Now modify the ode to make a Helmholtz Coil set up. Given the way the program has already been oded, this requires two very minor hanges: Set the solenoid s length equal to its radius, Redue the number of oils to just. Run the program and again note over how wide (in x, red plot) and how long (in y, yan plot) the field remains fairly onstant. Whih devie, the solenoid or the Helmholtz Coils, has proportionally large region of uniform field in its enter? Aording to the plot, what is the strength of the field in this region (you re really just looking at the y omponent, but the other two are negligible)? This should agree well with the value you d alulated using the theoretial relation. When you re satisfied with your program (always helps to hek with a neighboring group) save and upload a opy of it.
Phys 3 Lab 8 Ch 1 Interations with Magneti Fields 3 II. Charged Partile Moving in a Uniform Magneti Field a. Theory At its simplest, the magneti interation is the attration of parallel urrents and repulsion of anti-parallel urrents. As with the eletri interation, it s onvenient to use the field onept to divide the interation into two steps: 1) soure urrent (moving harges) produes field, ) field transmits fore to sensor urrent (moving harges). The fore transmitted by a magneti field to a moving harged partile is F qv B. () For simpliity and onreteness, we ll onsider a onstant and uniform magneti field pointing in what we ll all the y diretion and a harged partile initially moving in the x- z plane (so, with no omponent parallel to B.) It s a mathematial fat that the vetor that results from a ross produt ( F, in this ase) is perpendiular to the two vetors that are rossed ( v and B ). That leads us to three useful onlusions (that are baked up mathematially in Phys 33): a) Sine the fore must be perpendiular to B, no omponent of fore ats parallel to B and so the omponent of v that s parallel to B, that is, in the y diretion, remains onstant. If it s initially 0, it remains 0. b) Sine the fore is perpendiular to v, it annot speed or slow the partile, just hange its diretion. ) It follows that, sine the fore an t hange the magnitude of v and it an t hange the omponent of v parallel to B, it then an t hange the magnitude of the ross produt of v and B from a onstant F qvb. So the fore is of onstant magnitude and always perpendiular to B and to v, and the partile has onstant speed. This should sound familiar from Phys 31 as leading to uniform irular motion. Based on this reasoning, we know the rate of hange of momentum must have magnitude dp dt v m. R By the momentum priniple, this must equal the net fore, so v m qvb R v m qb R Two important preditions follow. First, for a given speed (whih must remain onstant) and onstant magneti field (and harge and mass), the radius of the partile s irular orbit is set at mv mv R. (3) qb qb
Phys 3 Lab 8 Ch 1 Interations with Magneti Fields 4 Seond, for uniform irular motion the speed is related to the angular frequeny of orbit by v R, so the frequeny with whih the partile orbits is qb qb, m m whih is virtually independent of v for v<< (where 1.) The orresponding period would be m T. (4) qb b. Qualitative Experiment At the bak of the lass room, an apparatus is set up for seeing eletrons pass through a fairly uniform magneti field. - + B R - e I I V In brief, a filament is heated and some eletrons get boiled off. A pair of apaitor plates surrounds the filament so these freed eletrons get aelerated by the field within the apaitor. By onsidering potential and kineti energies, the resulting speed of the eletrons relates to the apaitor s voltage through 1 1m mv ev, assuming v<< and that the initial veloity is negligible. In WebAssign, rephrase this expression to give the speed in terms of the voltage. v = (5) The eletrons pass through a small hole in one of the plates and now fly freely. All of this is within a glass bulb with a low density of gas when an eletron rashes into a gas atom, the atom exites and de-exites, and so radiates violet light whih illuminates the
Phys 3 Lab 8 Ch 1 Interations with Magneti Fields 5 eletrons trail. All of this is between Helmholtz Coils whih provide a fairly uniform magneti field (aording to Eq n 1) thanks to the urrent flowing through the oils. Pause and onsider: Looking at the illustration of the set-up, Use the right-hand rule for field prodution to first hek that the diretion of the field is appropriate given the indiated diretion of the urrent in the oils. Use the right-hand rule for the fore transmitted by a field to a moving harged partile (remember, the harge of the eletron is negative ) to hek that the diretion of the harge s defletion is appropriate when it s moving rightward at the bottom, upward at the right, leftward at the top, and downward at the left. For example: an out-of-the-page field would deflet a right-bound positive harge downward, and so deflets the negatively-harged eletron upward. Similarly, hek that the defletion of the eletrons at these four points is onsistent with the rule that like urrents attrat and opposite urrents repel. For example, when the negatively-harged eletron is moving right, the rightbound urrent at the bottom of the loop repels it and the left-bound urrent at the top attrats it. In WebAssign, ombine equations 1, 3, and 5 (assuming v<< so 1) to give an equation for the radius of eletrons path in terms of the urrent through the Helmholtz oils and the voltage aross the apaitor plates (and other relevant fators like the oils radius, R ) rather than v or B. Note: the q in Eq n 3 would be e for our eletrons. R= (6) So the violet eletron trail should urve with a radius aording to Eq n 6. With the experimental apparatus at the bak of the room, Vary the aelerating voltage (dial on the power supply, keep in the 150V 300 V range) and see the radius vary in aordane with Eq n 6. Vary the urrent (dial at Helmholtz Coils base) through the Helmholtz oils (and thus the magneti field), and see the radius vary in aordane with Eq n 6.. Simulation You ll write a simulation to model an eletron s motion through a uniform magneti field. You ould modify your Helmholz.py program to inlude a moving eletron; however, the repeated realulation of magneti field everywhere the eletron goes would be unneessarily omputationally-intensive / slow. Sine your Helmholtz.py program has demonstrated that oils an produe a uniform field over a broad region, you ll simply write a program that takes a uniform field as a given. This is a good opportunity to refresh your memory of how to write a whole new program (rather than inrementally modifying an existing one). Create a new program in VPython, and name it EletronInB.py.
Phys 3 Lab 8 Ch 1 Interations with Magneti Fields 6 What follows will guide you through filling in the setions of the ode. Beginning As with all of your programs, you ll want to start off with the two from import lines (the exat same ones that start your Helmholtz.py program) and follow those with a omment line that inludes your names. Constants Define: the magneti field vetor (B) to be 0.0001 Tesla in the +y diretion a variable (Bsale) to sale the size of the arrows representing the magneti field (start with 500.0) 10 the time step (dt) to be 1 10 s the maximum time (tmax) to be 1 10-6 s. Objets Create a sphere alled eletron to represent an eletron by filling in the definition eletron = sphere(...) with the following attributes o an initial position (pos) of 0,0,0 m. o a radius of 5 mm (muh larger than its atual size so it an be seen in your simulation). o a olor of olor.blue 31 o a mass of 9.1 10 kg 19 o a harge of 1.610 C o an initial veloity of v = vetor(0,0,1e6), units of m/s o and set it to leave a trail everywhere it goes by inluding, along with these other attributes, make_trail=true inside the definition of the eletron. At a few observation loations, reate arrows to represent the uniform magneti field. You an do this however you like, but here s one way: d = 0.1 #in meters, oordinates of observation loations obslos = [vetor(-d,0,-d), vetor(-d,0,d), vetor(d,0,-d), vetor(d,0,d)] #list of obslos for obslo in obslos: arrow(pos = obslo, axis = B*Bsale, olor = olor.yan) Initial Values Initialize the time at t = 0. Calulations It s been a while sine you ve written dynamis programs (ones that simulate the motion of partiles in response to fores), so here s a quik refresher: You have a while loop that steps through time, from t = 0 to some final time, tmax; inside that loop, you realulate the fore on the partile (aording to Eq n in this ase), then the partile s new veloity aording to the approximation F v v new old t m,
Phys 3 Lab 8 Ch 1 Interations with Magneti Fields 7 and then the partile s new position aording to r r vt Of ourse, you update the time too; new old. t new t old t I ll get you started, but you ll need to omplete eah line of ode (remember, A B in Python is ross(a,b)). While t < tmax: F = eletron.v = eletron.pos = t = rate(50000) Run the program and make sure the eletron behaves as you expet. Pause and onsider: How would you expet the harged partile to behave differently if it had a positive harge instead of a negative harge? Flip the sign of the harge and run the program again. How would you expet the harged partile to behave differently if it were initially going faster? Double its initial speed and run the program again. How would you expet the harged partile to behave differently if it initially had a omponent of veloity parallel to the field (remember, there an be no omponent of the fore parallel to the field)? Give the partile s veloity an initial y omponent of 1 10 4 m/s and run the program again. Compare with Theory: Radius. Aording to Eq n (3), we expet a speifi radius of the partile s orbit for given mass, speed, harge, and magneti field. You ll have the program determine the theoretial and simulated radii. Theoretial Just before and outside the while loop, add a line to alulate the expeted radius, rtheory, based on the harge, mass, field, and speed (whih is mag(eletron.v)). Follow that with the print line print( the expeted orbital radius is, rtheory, m. ) What value do you get?
Phys 3 Lab 8 Ch 1 Interations with Magneti Fields 8 Simulated Assuming the eletron is indeed exeuting irular motion in the x-z plane, then the diameter of the orbit would the its maximum z oordinate minus its minimum z oordinate (and the radius would simply be half the diameter). So, To find the maximum and minimum x oordinates, add following lines o In the Initialization setion, define Zmax = eletron.z Zmin = eletron.z o o o At the bottom inside the while loop add if eletron.z > Zmax: Zmax = eletron.z Add some similar lines to determine Zmin. At the end of the program (outside the while loop), add (and omplete) the line rsimulate = #the radius of the simulated orbit, based on Zmin and Zmax print( the simulated orbital radius is, rsimulate, m. ) Run the program and see how the theoretial and simulated radii ompare. Period Aording to Eq n (4), we expet the period of the eletron s orbit to depend on the field strength, speed, mass, and harge in a speifi way. Theoretial Before and outside the while loop, add a line to alulate the expeted period, Ttheory, based on the harge, mass, and field strength. Then add a print statement to print out this value. What value do you get? Simulation As the partile goes bak and forth, the x-omponent of its veloity flips sign twie per orbit first swithing from heading left to heading right, and then swithing from heading right to heading left. So you an determine the period by ounting the time between these flips, and doubling it. In the initialization setion, define Fliptime = 0 #will keep trak of when the veloity flips diretion At the top inside the while loop add the line
Phys 3 Lab 8 Ch 1 Interations with Magneti Fields 9 Oldvx = eletron.v.x At the bottom inside the while loop add the lines if Oldvx/eletron.v.x <0: #true if veloity flipped diretion T = *(t Fliptime) Fliptime = t At the bottom of the program, add the print line print( the simulated orbital period is,t, s. ) Run the program and see how the theoretial and simulated periods ompare. When satisfied, save and upload the program. III. Dipole in a Uniform Magneti Field a. Theory A dis magnet is the most readily identifiable magneti dipole. As the name suggests, a magneti dipole has two poles North from whih magneti field lines emanate, and South into whih magneti field lines terminate. (In reality, magneti field lines are losed loops that ontinue on through the body of the magnet). The simplest magneti dipole to piture and understand is a single loop of urrent harged partiles speeding around in a irle. From Phys 31, you may reall that angular momentum an be defined for partiles moving around in a loop, and that property is useful for disussing the work and torque required to hange the partiles irular motion. As the angular momentum fouses on the motion of mass, the magneti moment fouses on the orresponding motion of harge. It is similarly useful for disussing the work and torque assoiated with the moving harges interations with a magneti field. Like angular momentum, the magneti moment vetor s diretion follows from a righthand rule: if the urrent flows ounter-lokwise about the y axis, then the magneti moment points in the +y diretion; if the urrent flows lokwise about the y axis, then the magneti moment points in the y diretion. The torque exerted via a magneti field on a magneti dipole is B. Applying a right-hand rule to this ross produt makes it lear that the torque must be perpendiular to both the magneti moment and the field. Of ourse, the diretion of the torque is the diretion of the axis about whih the dipole is getting twisted. An example of this situation is illustrated below. B
Phys 3 Lab 8 Ch 1 Interations with Magneti Fields 10 In this example, the magneti moment of the small oil points up and right while the field of the large oils points up; aordingly, the torque vetor points out of the page, meaning that the small oils is pushed so its right edge would rise and its left edge would fall. Though mathematially more omplex, a oneptually simpler argument is that the small oil experienes a torque to bring its urrent into better alignment with the urrent in the large oils beause parallel urrents attrat and anti-parallel urrents repel. b. Qualitative Experiment Experimental Setup A magneti dipole (dis magnet) hangs from a spring and is mounted so it an pivot in response to a torque transmitted by a magneti field. This hangs near the middle of a pair of Helmholtz Coils whose urrent, and thus field, you an ontrol with the power supply. (It s for a later experiment that the urrent is routed through a preision resistor.) Torque Experiment Make sure the magnet s dipole moment points at an angle off vertial (an arrow on its side marks its orientation.) If it isn t already, you an do this by lifting the plasti ap, (along with rod, spring and dipole) from the graduated ylinder, tipping the dipole, and then replaing the ap (rod, spring, and dipole.) Turn on the power supply (swith is on the bak) and observe the dipole s response. Turn off the power supply, unplug the ables form it and re-plug them reversed (red able to blak port and vie versa). This reverses the diretion of urrent flow through the Helmholtz oils, and thus the diretion of their magneti field. Turn the power supply bak on and observe the dipole s response. Fore Experiment Dial up the urrent flowing through the oils, thus the field strength, and observe the dipole s response. Though the intrinsi urrent in the dipole (in this ase, eletrons orbiting their atoms) is attrated to the parallel urrent flowing through the Helmholtz oils, the attration to the top oils is the same as that to the bottom oil, so the dipole experienes no net fore under the symmetri, onstant-field ondition.
Phys 3 Lab 8 Ch 1 Interations with Magneti Fields 11 Interation with a linearly-varying Magneti Field All of the preeding theory, simulations, and experiments have pertained to uniform magneti fields. Now you ll explore the simplest non-uniform magneti field one that points only along one axis with a strength (and diretion) that varies linearly along that diretion. I. Produing a linearly-varying Magneti Field a. Theory Before experimenting with or simulating a linearly-varying magneti field, we ll digress and onsider how one produes suh a field. As you might expet, this ould be ahieved by having a solenoid of linearly varying urrent (for example, the top rung aries 1 amp lokwise, the bottom rung aries 1 amp ounter-lokwise, and the urrent through the in-between rungs smoothly varies the one to the other as you move along the length.) However, that is both tehnially diffiult and unneessary. As a pair of Helmholtz oils an produe a rather uniform field between them, oils in an anti- Helmholtz onfiguration (opposite urrents in the two oils) produe a rather linearlyvarying field between them. The expression for the field along the axis between two oils in anti-helmholtz onfiguration (one entered at y = -R / and one entered at y = +R / with opposite urrent) is B y o y 1 R IR R 3/ o I R 1 y R R 3/. So this field varies along the axis like db 1 y IR y R o 3 5/ dy 1 y R R o 3 1 I R y R 1 y R R 5/ At a loation fairly near the midpoint, y << R, so the y in the denominator is negligible, and the y dependene in the numerator then anels out, leaving us with db dy y I I 3 (7) o 0. 859 R 5/ 4 5/ o R So the slope of the field vs. position along axis should be onstant, that is, the field should vary linearly along the axis (in the y<<r region.) b. Simulation To get a feel for this field, you an easily modify your Helmholtz.py program to simulate the anti-helmholtz onfiguration. Open your opy of Helmholtz.py In the line where Idl is defined, insert a fator of *y/l. This will simply equal 1 for the urrent loop up at above y = L/, but it will equal -1 for the urrent loop down at y = -L/, thus flipping the diretion of the urrent. (If you re interested in.
Phys 3 Lab 8 Ch 1 Interations with Magneti Fields 1 seeing what a linearly-varying solenoid would be like, just inrease the number of urrent loops to something large, like 50.) Run the program and observe the length of the region over whih B y varies linearly with y. II. Dipole Interating with a Linearly-Varying Magneti Field a. Theory Imagine a dipole built of a oil of urrent-arrying wire, and the oil has nonnegligible height. In a regular Helmholtz oil setup, the top loop of this dipole is just as strongly attrated to the urrent in the upper Helmholtz oil above it as the bottom loop of this dipole is attrated to the lower Helmholtz oil below it, so the dipole experienes no net fore. Equivalently, we an relate this to the assoiated uniform field (rather than diretly to the urrents that produe it) and say that in the uniform field of the Helmholtz oils, there is no net fore on a dipole. You ve already observed that a dipole will twist but not aelerate and translate due to a uniform field. However, in an anti-helmholtz setup, if the oil below attrats the dipole, the oil above repels it, so the dipole does feel a net fore. Equivalently, we an say that if the field varies along the length of the dipole, then the two faes feel unbalaned fores, and so there is a net fore. Mathematially, that is expressed as F B, In our ase, the dipole is free to (and will) align with the field that points along the y axis, so we an write more speifially and simply F y dby. (8) dy b. Qualitative Experiment Experimental Setup This is the same as for the Helmholtz onfiguration exept the power supply is onneted so the urrent flows in the opposite diretions in the two oils. Note the different wiring in the diagram.
Phys 3 Lab 8 Ch 1 Interations with Magneti Fields 13 Experiment Turn on the power supply and pass a urrent of about amps. In the plasti ap, loosen the set srew that holds the brass rod (to whih the spring and thus magnet are attahed) and raise it enough that the dipole goes above the top oil, then lower it until the dipole is below the midpoint. What does the dipole do as it rosses below the midpoint? Pause and onsider: Thinking of the field illustrated in your simulation, reason out why the dipole behaves as it does as you lower it. Here s a start: when the dipole is above the top oil, it s oriented so that it s aligned with and attrated to the urrent in the top ring. As you lower the dipole. Quantitative Experiment You ll determine the magneti moment of the dis magnet by varying the urrent, and thus derivative of the magneti field and observe the orresponding streth of the spring, and thus fore on the dipole. Together, the fore and derivative of the field determine the dipole moment (aording to Eq n 8.) If it s on, dial down the urrent and turn off the power supply. Adjust the height of the dipole so the top of its holder is aligned with the 0 mark on the plasti tube that surrounds it. Turn on the power supply. For urrent values from 0.5 amps to 3.0 amps, note the height of the top of the dipole s holder. Enter the urrent values and displaements into the table in WebAssign. o To get more preise measurements of the urrent, a multi-meter is monitoring the voltage aross a preision 1 resistor; by Ohm s law, the urrent in amps equals the voltage in volts. Given that our oils have a radius of 0.07m and are atually 168 loops of wire, Eq n 7 an be made more speifi to our set up as db y 0.037 I T/(m A) dy. Use this to fill in the orresponding olumn of the table. The spring from whih the dipole hangs has a stiffness of k s = 0.01 N/m. Use this to fill in the fore olumn of the table. Plot fore as a funtion of the field s derivative, and from the slope, determine the magnet s magneti dipole.
Phys 3 Lab 8 Ch 1 Interations with Magneti Fields 14 Helmholtz.py
Phys 3 Lab 8 Ch 1 Interations with Magneti Fields 15 EletronInB.py The modifiation to Helmholtz.py to make it antihelmholt.py