184_projects:s20_project_9_sol

You've just received a phone call from the head of the Wildlife Tracking Foundation of Lakeview. Dr. Taulest reports that they have recently had an unprecedented disappearance of their yellow-bellied canaries that like to roost in the cliffs overlooking the lake. Their most popular nesting area on the eastern edge of the cliffs is completely empty. It's almost like they decided to migrate 4 months ahead of schedule! The Wildlife Tracking Foundation is also reporting that the fish, especially those near the north-eastern bank of the lake, have suffered a sudden decrease in their population. Dr. Taulest claims that it is all your fault since you had to build a large power line to run the Hawkion detector. Clearly, the magnetic field from the power line interfered with the yellow-bellied canaries (who rely on Earth's magnetic field for migration purposes), causing their vacation from the vicinity.

As the lead engineers on the Hawkion project, your boss has kindly asked that you disprove Dr. Taulest's claims. The company could be facing ridiculous litigation fees if the Wildlife Tracking Foundation has a case against you. To start, you go back and consult the handy-dandy blueprints you drew up when designing the transmission line.

This problem can be as complicated as you want to make it. This solution will be the most simple solution, with a list at the end of ways you can make it more challenging.

Learning Goals

  • Use the right hand rule to determine the direction of the magnetic field near a wire.
  • Calculate the magnetic field from a wire segment with a current and explain the role of superposition in your solution.
  • Explain what the pieces of the integral are ($d\vec{l}$, $\vec{r}$, and limits) and how you determined them for your solution.
  • Relate the magnetic field from a current to the magnetic field from a single moving charge - explain how these are similar and/or different.

Learning Issues

  • You may want to remind your students that in this class they are only expected to setup the integral. There is no expectation for this class that they be able to solve an integral. Any instances where they need to solve an integral to get some kind of a numerical value, they will have access to resources to do it for them (e.g. WolframAlpha).
  • Have students draw their representations for $\vec{r}$, $dx$, $x$, etc. There are a lot of terms to juggle in this problem, and being explicit about drawing these out and labeling them can be really helpful to students.
  • This is integral fun day - make sure that students understand what they are doing. Watch out for groups going into crazy math land without any reasoning behind their work.
  • Likely there will be a lot of confusion about the $\vec{dl}$ and $\vec{r}$ as these are the hardest pieces to figure out. Grounding in the physical reasoning can help here (i.e. $\vec{dl}$ is a little bit of length of the current carrying wire) and relating to what we did with electric fields (i.e. what we did before, we split up the charge and called the chunks $dq$ - what is $\vec{dl}$ then?) can be very helpful here.
  • Watch the RHR - might be confusion between the $\vec{B} \propto \vec{dl} \propto \vec{r}$ and $\vec{F} \propto \vec{v} \propto \vec{B}$ ones. You can also go with the RHR shortcut for $\vec{B}/\vec{I}$ in this problem since it's a current carrying wire. There are a lot of variations with the RHR and students may start getting tripped us when figuring out which version to use and when.

Timing Things

  • If your groups have not settled on the corner as their observation point after 30 minutes, push them to use that point. Ask them about the “simplest solution” (one where two out of the three wires contribute no magnetic field). If there is time at the end, they can do a more complicated observation point(s), but it is not worth it to start with a complicated scenario.

The first step here is to choose the observation point that you want to calculate the magnetic field at. The easiest point is to pick the point on the cliffs that is in the north-east corner of the lake (following the information given in the text of the problem). In doing this, you also want to make a couple of assumptions:

  1. The observation point is at the same height as the power line (aka, no z-direction needed)
  2. The current is constant/steady-state
  3. The observation location is very close to the where the power line bends (and we will approximate it as at that vertex)
  4. We also need to choose a coordinate system, more specifically, where are we going to put the origin in defining the positions of all of the chunks of current and observation locations. This is a very important step any time we are working with vector quantities.
  • Question: Where did you pick your observation point? Why did you pick it there? (Is there a more simple observation point that you can choose?)
  • Expected Answer: We picked the cliffs near the north-east side of the lake because that's where most of the strange animal observations were occurring. This simplified the problem because the contributions from 2 of the 3 wires were zero at this point. This is because the dl and r vector point in the same direction for these wires so the cross product is zero.

To find the magnetic field at this point, you want to split the wire into three chunks, one for each segment of the power line. Since we have a current-carrying wire, we want to use the following version of the magnetic field equation to find the B-field for each chunk:

$$\vec{B}_{wire}= \frac{\mu_0 \cdot I}{4 \pi} \int \frac{d\vec{l}\times \vec{r}}{r^3}$$

  • Question: What is $d\vec{l}$ in this equation?
  • Expected Answer: It is a little bit of length of the wire. It's direction points in the same direction as the conventional current.
  • Question: What is $\vec{r}$ in this equation?
  • Expected Answer: It is the separation vector - same as uj, it points from the source (the dl or chunk of wire) to the observation point.
  • Question: What do the limits on this integral tell you about?
  • Expected Answer: The limits tell you how long the wire is with the current in it. (This is because we are integrating over dl)

For the first and second chunks, the $d\vec{l}$ (which follows the direction of the current) would always be parallel to the $\vec{r}$. This means that the cross product $d\vec{l} \times \vec{r}$ = 0 for both of these segments. So we only really need to find the magnetic field from the 3rd chunk of wire.

In this case, we pick the origin to be at the right end of the wire. $d\vec{l}$ then is a little piece of that wire and points in the same direction as the current, so depending on how students pick their coordinates: $$d\vec{l} = dx \hat{x}$$ As per normal, the $\vec{r}$ should point from the little piece of wire to the observation point. In this case, you have to use some trig to find the distance away from the end of the wire. The $\vec{r}$ is then equal to: $$\vec{r}= \langle x+1000\cdot cos(60), 1000sin(60), 0 \rangle$$ With a magnitude of: $$r=\sqrt{(x+1000\cdot cos(60))^2+(1000sin(60))^2}$$

  • Question: Why is there an 'x' in the r-vector and not a 'dx'?
  • Answer: Because the r-vector is always going to have a 'large' distance in the x. The 'dx' means a little bit of length in the x (or an infinitesimal length) - so it wouldn't make sense to have it in the r-vector.

When you take this cross product (either by hand or with wolfram), you get: $$d\vec{l} \times \vec{r}=1000sin(60)dx \hat{z}$$ Note that this is only the y part of the $\vec{r}$ - which should make sense as you want the part of the vector that is perpendicular to $d\vec{l}$. Also because of how we defined our coordinates, $+\hat{z}$ points into the page.

At this point you just plug everything into the integral with bounds from 0 to 2000 (that is where your wire starts and stops): $$\vec{B}_{wire}= \frac{\mu_0 \cdot 2.5}{4 \pi} \int_0^{2000} \frac{1000sin(60)dx}{\sqrt{(x+1000\cdot cos(60))^2+(1000sin(60))^2}^3}\hat{z}$$

  • Question: How does your answer compare to the magnitude of Earth's magnetic field?
  • Expected Answer: See below

You should find that the B-field is equal to $\vec{B}_{wire}=1.2*10^{-10} T$ in the $+\hat{z}$ direction or into the page. This is 1000 times smaller than Earth's magnetic field (which students should be able to look up - ~50 $\mu$T) and therefore is likely not the cause of the disappearing birds/fish. I wonder what it could be??? Bum, bum, bum!!

If students finish this early, you can push them to find the magnetic field at a different point on the cliff (mid-lake) which means you would have to set up a similar integral for Wires 2 and 3. Or you can ask them for the magnetic field on the ground - which means you have to have a z-direction in the r-vector.

Discussion Prompts

  • Question: How did you pick your $d\vec{l}$, your $\vec{r}$, and your limits on the integral?
  • Expected Answer: $d\vec{l}$ is a little bit of length that points in the direction of the current - in this case was $-dx\hat{x}$. $\vec{r}$ points from the little bit of $d\vec{l}$ to the point of interest - same as from before source to observation point. The limits on the integral should be from 0 to 2000 because you want to add up the bits along the 3rd bit of the wire.
  • Question: How did you use superposition in your solution? Or why did we need to use an integral in this case?
  • Expected Answer: Because a current is many moving charges, each of which contributes to the magnetic field at the observation point. We want to add up all of those contributions, so we need to use an integral. This is where the superposition happens in our solution. (Superposition can also appear if they used all 3 wires in their solution.)
  • Question: How does this process relate to what we did to find the magnetic field from a single moving charge?
  • Answer: It is a very similar process only now we have I*dl instead of q*v and we have to do an integral to add up all the lengths of wire instead of just calculating a single charge.

Evaluation Questions

  • Question: What direction does your magnetic field point? Does that make sense? How do you know if that is right?
  • Expected Answer: It points into the page (or down towards the ground) - this agrees with the Right Hand Rule. Have them show you the right hand rule - if they seem to be struggling, have them do the RHR at several points.
  • Question: How did you simplify your model in this case? What would change about your solution if you didn't make those assumptions?
  • Expected Answer: The observation point is at the same height as the power line (aka, no z-direction needed), the current is constant/steady-state, the observation location is very close to the where the power line bends (and we will approximate it as at that vertex)
  • Question: If this were a group exam, how would you evaluate your answer? What are the limitations of this model? What would make it better?
  • Expected Answer: This is a very simplified solution - picking any other observation point would be more accurate (especially with regards to the birds' locations). We could account for the height of the cliff and all three wires in that case, which would improve our model.

Extension Questions

  • Question: What is the magnetic field 1 cm above the observation point? Would you expect it to be bigger or smaller than your previous answer?
  • Expected Answer: It should be bigger as now there are contributions from all three wires, not just one. Plus the r vector will be very small for 2/3 of the wires, so it should be much bigger than at the previous observation point. You can have them actually calculate it if there is time.
  • Question: What would change if the observation point was on the ground (i.e. for the fish) rather than at the 5.5 m?
  • Expected Answer: The r-vector would then have a z-component, which would change the direction of the magnetic field. You can walk through the right hand rule to figure it out or have them calculate it if there's time.
  • Do you have any lingering questions? Anything that doesn't make sense or that you would want to go over again?
  • Expected Answer: TBD

Since you are investigating the mysteries of Lakeview and the surrounding area, you decide to take some time to walk into town and get to know some members of the community. As you are strolling around town, you hear a holler. “Aya!!! Hey, howdy!! Come check out this ol' piece of junk I got right here, y'all!” It's Dr. Daryn McPaddel, the town inventor/engineer who is always showing her neighbors strange contraptions that she has made, and also happens to live adjacent to Lakeview's scrapyard, which allows her to tinker and invent to her heart's delight. This is what you see:

All the commotion around Dr. McPaddel's magnetic crane-thing has caused some neighbors to gather and see what all the fuss is about. Some of them approach you as they scratch their heads in disbelief. “Hey, you know about electricity and magnets and whatnot, can you explain what's going on?” You agree to explain in detail how the thing works since you want to form a bond with the townsfolk. The best way would probably be with a visual, and Dr. McPaddel has already created a visual of the coil of wire inside the contraption. All that's left to do is show what the magnetic field looks like.

xaxis = cylinder(pos = vec(-3, 0, 0), axis = vec(6, 0, 0), radius = 0.01, color = color.white)
yaxis = cylinder(pos = vec(0, -3, 0), axis = vec(0, 6, 0), radius = 0.01, color = color.white)
zaxis = cylinder(pos = vec(0, 0, -3), axis = vec(0, 0, 6), radius = 0.01, color = color.white)
 
## Setting up the constants
mu0 = 4 * pi * 10 ** -7    # magnetic constant in standard units
I = 5000                   # current in ring
R = 1                      # radius of ring
thickness = 0.05           # thickness of ring
 
## Drawing the ring
ring = ring(pos = vec(0, 0, 0), axis = vec(0, 0, 1), size = vec(thickness, 2 * R, 2 * R), color = color.blue)
 
## Splitting the ring into little pieces
N = 100                    # number of little pieces
dtheta = 2 * pi / N        # angle between adjacent little pieces
theta = 0
little_pieces = []
 
## Determining the location of each little piece and determining dl
while theta < 2 * pi:
    r_source = vec(0, 0, 0)
    dl = vec(0, 0, 0)
 
    little_piece = arrow(pos = r_source, axis = dl, color = color.green)
    little_pieces.append(little_piece)
 
    theta = theta + dtheta
 
## Picking some points to observe the magnetic field
observation_radii = [0, 0.4, 0.8, 1.2, 1.6]
observation_angles = [0, pi/2, pi, 3*pi/2]
observation_heights = [0]
r_obs_vectors = []
 
## Putting the observation points into a list
for rad in observation_radii:
    for angle in observation_angles:
        for height in observation_heights:
            r_obs = vec(rad * cos(angle), rad * sin(angle), height)
            r_obs_vectors.append(r_obs)
 
## Visualizing the magnetic field
for r_obs in r_obs_vectors:
    B_total = vec(0, 0, 0)
    for little_piece in little_pieces:
        r_source = little_piece.pos
        dl = little_piece.axis
 
        r_sep = vec(0, 0, 0)
        B_little_piece = vec(0, 0, 0)
 
        B_total = B_total + B_little_piece
    arrow(pos = r_obs, axis = B_total, color = color.yellow)

Learning Goals

  • Practice the right hand rule, and make predictions for what magnetic field looks like from a ring of current
  • Relate superposition in the code to how an integral is constructed
  • Investigate the physical meaning of how an integral splits up a wire into “little pieces”
  • Practice setting up an integral for a different shape than a straight line
  • Gain some experience working in cylindrical coordinates

Learning Issues

  • This problem has a comparatively large amount of code to digest. This may intimidating for some students. Encourage them to take some time at the beginning of class and talk through the code. Commenting the code can be very effective at trying to understand what each line is doing. This process can also help students to identify where they need to change/add code.
  • The mathematical representation for $\vec{dl}$ and $\vec{r}$ come from using polar coordinates. If students are unfamiliar with using polar coordinates, then this may be challenging for them. If students are stuck at about an hour into class, start pushing them to try doing some Google searches for polar coordinates, representing a tangent vector in terms of polar coordinates, etc.
  • The observation points are contained in a list. This means that there are going to be a lot of nested loops in the code (calculating the magnetic field at a bunch of locations due to a bunch of little pieces of current). The problem should serve as good practice for working with nested loops, but keep an eye on your groups to see if they are getting lost in all of the loops. Having them comment “##Looping over observation location” at the beginning of the loop for locations, in this example, and then commenting “##End of observation location loop” at the end of the loop can help keep these straight.
  • The plan laid out in this solution is very thorough. Make sure that your students are planning. There is a lot of temptation in this problem to just fumble around with sine's and cosine's until the code works, so encouraging the use of the quadrants, planning, and drawing out representations is going to need to be a focal point.

What follows is a possible solution. Students may approach it differently. The code provided does put some railings on the path students might take to construct a visual of the magnetic field, so hopefully this solution is not too far off, and can serve as a guide. Starting off should be pretty straightforward. Students are meant to produce a visual of the magnetic field – they have done this before. The trickiest part is that they are now working in cylindrical coordinates. The main reason I like having the students do this problem in cylindrical coordinates is because it forces them to consider the physical significance of $\text{d}\vec{l}$ – it even shows up in the visual once they figure stuff out.

Goal: The goal of the project is to produce a visual of the magnetic field near the ring. Students will probably figure this out shortly after reading the problem and running the minimally working code. The tough part is forming a plan.

Plan: These are the steps students might be motivated to take in order to reach the goal.

  1. Make a representation of the ring.
  2. Select a little piece of the ring.
  3. Determine the location of the little piece, and recognize that the location is what we sometimes call $\vec{r}_{source}$. Put that into the code.
  4. Determine $\text{d}\vec{l}$ of the little piece, and put that into the code.
  5. The observation locations are given.
  6. Determine the separation vector for a given source and observation, and put that into the code.
  7. Use the separation vector and $\text{d}\vec{l}$ to find the magnetic field from the little piece, and put that into the code.
  8. Superposition is already set up for them, so at this point, the visual should be working. Student will likely need to add a scaling factor.
  9. Students can toggle/add observation points as desired.

Here is a fully working code. The plan is fleshed out in detail after the code.

Step 1: Make a representation of the ring, and make a prediction for the magnetic field Students will likely adopt this representation right away, since this is what is in the code. There are a lot of approximations and assumptions associated with using this representation.

  • Question: What are some approximations and assumptions you need to make in order to use that representation? (The ring.)
  • Answer: Here are some possible assumptions and approximations:
    • There actually is a loop of wire inside the magnet crane (Assumption)
    • The ring of wire is perfectly circular (Approximation)
    • The wire running along the crane arm to and from the ring does not contribute significantly to the magnetic field (Approximation)

As far as making a prediction goes, this will involve some right hand rule finagling.

Step 2: Select a little piece of the ring. This is easy to draw, but difficult to put into the code. Students might need to draw this one on the whiteboard in order to

Step 3: Determine the location of the little piece, and recognize that the location is what we sometimes call $\vec{r}_{source}$. Put that into the code. It might help to be explicit that location is defined with respect to the origin, and so yes location can be a vector.

  • Question: What is the source of magnetic field?
  • Answer: Current!
  • Question: How is the “little piece” related to $\vec{r}_{source}$?
  • Answer: Each little piece has its own location, or $\vec{r}_{source}$. This is a little tricky, since location is a vector, and students might be used to thinking of $\text{d}\vec{l}$ as the “vector” associated with little pieces.

Step 4: Determine $\text{d}\vec{l}$ of the little piece, and put that into the code. Students might create the following visuals, or something similar, to figure out the direction and magnitude of $\text{d}\vec{l}$. Another way to find the length of $\text{d}\vec{l}$ is to see that the circle is split into N=100 pieces, and then just divide the circumference by N.

  • Question: What is the magnitude of $\text{d}\vec{l}$?
  • Answer: It is the length of a little piece of the ring.
  • Question: What is the direction of $\text{d}\vec{l}$?
  • Answer: It is directed along the current in the wire.

Step 5: The observation locations are given. Students might figure this out by reading the code.

Step 6: Determine the separation vector for a given source and observation, and put that into the code.

  • Question: What is the separation vector?
  • Answer: Source to Observation!
  • Question: What/where is the source of magnetic field?
  • Answer: Current. But we don't have a “location” of current unless we have the little pieces. So, each little piece is a source of magnetic field.
  • Question: Where is the observation point?
  • Answer: We have many observation points… So it could be anywhere. The tricky part to be clear about is that when we look at one observation point, we have to look at all the little pieces before moving on to the next observation point. When we apply superposition of magnetic fields from all the little pieces, this is all for ONE observation point.

Step 7: Use the separation vector and $\text{d}\vec{l}$ to find the magnetic field from the little piece, and put that into the code. If students have done everything in Steps 1-6, this is just a matter of putting Biot-Savart Law into the code.

Step 8: Superposition is already set up for them, so at this point, the visual should be working. Student will likely need to add a scaling factor. The way the sample solution is written, $5\cdot 10^7$ seems to be a good scale.

Step 9: Students can toggle/add observation points as desired.

Discussion/Evaluation Questions

  • Question: Comment the code that's given.
  • Answer: This is a good exercise if students are stuck. There is a lot of code given to them and understanding it might help them figure out what they need to change.
  • Question: How does the magnitude of the magnetic field compare at 0.8 meters and 1.2 meters?
  • Question: It is significantly bigger at 0.8. If students split the ring in half, they can compare the direction of the magentic field contributions from each half, it's kind of cool how they differ. Visual below.
  • Question: How does the visual help explain how the crane works?
  • Answer: Metal objects are magnetized by the field. Then what? The mechanism by which the upward force is produced is a great mystery. (How can the force be in the same direction as the magnetic field???) The visual helps us see the field, but the upwards force that the objects experience is a little out of reach. Remember – magnetic force is always perpendicular to magnetic field.
  • Question: How can the visual be improved?
  • Answer: One thing to do is add observation points by adding to the list observation_heights. Also, students may have other ideas. This might involved circling back to the discussion on assumptions and approximations.
  • Question: How can you check that the code is producing the correct output?
  • Answer: There are two things I can think of. Students might have more. These could also serve sort of like extension questions.
    • The shape of the field should match the prediction. If students didn't make a prediction, you could have do a post-coding exercise with a right hand rule to verify that the direction makes sense.
    • Students could print the magnetic field at an observation point to see how the numbers look. They could set up an integral to see if the numbers are the same if they do one observation point by hand.
  • Question: Does the magnitude of the magnetic field make sense?
  • Answer: Hopefully. If students print out values of the magnetic field, they should be getting numbers up in the couple-of-milli-Teslas range, which is much bigger than the Earth's magnetic field, which makes sense.

Extension Questions

  • Question: Check that a numerical output for one of your observation points makes sense.
  • Answer: Students could actually set up an integral here and put into Wolfram Alpha. It probably makes sense to pick a point somewhere along the axis of the ring. There is also an example in the notes that might help students do this.
  • Question: Tie each of your edits in the code to the setup of an integral.
  • Answer: This could involve commenting the code, or just doing the setup on the whiteboard, like in the previous question.
  • Question: Are there additional questions?
  • Answer: Students might have questions. If not, you can bring up the integral setup, and see if they have questions on how to setup the integral, and what each part of the integral means physically.
  • 184_projects/s20_project_9_sol.txt
  • Last modified: 2019/12/13 00:21
  • by dmcpadden