184_projects:s18_project_13_sol

Thanks to the research that folks at the Hawkion were able to do on the EM-boar tigers, the people of Lakeview have been able to set up a scientifically supported protection system. They have installed a giant magnet into the cliffs nearby, and it is aligned in such a way that there is a constant (practically speaking) $5 \text{ mT}$ magnetic field directed into the ground near Lakeview and the surrounding area. The hope is that the field will ward off the EM-boar tigers and keep the people of Lakeview safe.

However, there is a dilemma that has emerged. Local inventor and superstitious person Manny Callabero is starting to create a fuss around town because he believes the magnetic field is going to destroy his car. He has been driving around with a voltmeter attached to his engine and he claims there is a potential difference across the engine when he's driving fast. Once, he said, there was a potential difference of $0.15 \text{ V}$. He is worried that if the voltage across his engine reaches much higher, it will surely be destroyed and he could end up in danger. Out of the goodness of your heart, you wish to assuage Manny's worries and explain to him why the potential difference exists. You should also provide some guidelines for Manny to make sure he doesn't cause the potential difference to exceed $0.15 \text{ V}$.

Manny's Engine

Students should be able to reduce this problem into a rectangular prism (the engine) moving through a constant magnetic field. One way to do the problem is to put the $x$-axis pointing west to east, and the $y$-axis pointing south to north. We get the following diagram:

Representation

Assumptions:

  • The engine can be modeled as a uniform rectangular prism.
  • The engine can be assumed to be 1 meter wide.
  • The motion of the engine is perfectly perpendicular to the B-field.
  • The B-field is constant.
  • The mobile charge carriers in the engine are electrons.
  • The engine is neutrally charged.
  • The engine is not directly connected to another conductor and does not lose or gain any charge.

As the engine moves through the B-field, there is a magnetic force on the mobile electrons in the engine: $$\vec{F}_B=q\vec{v}\times\vec{B} = -e \cdot (v \hat{x}) \times (-B \hat{z}) = -evB \hat{y}$$

Note: the sign of the force may flip depending on how they orient the engine and what they choose as the mobile charge carriers in the engine.

The electrons will be forced to the bottom of the engine in our diagram, or to the right from the driver's perspective. But this will stop at some point, because the shifting charges create a potential difference that opposes the motion of additional mobile charges. Eventually, charges will stop moving towards the bottom (or to the right) when the net force is zero. This requires: $$\vec{F}_E = -\vec{F}_B = evB \hat{y}$$

Students may wish to find electric field: $$\vec{E} = \frac{\vec{F}_E}{q} = -vB \hat{y}$$

Or they may skip straight to the potential difference: $$\Delta V = \left| \vec{E} \right| L = vBL$$

The following visual may be helpful: Representation

If the students assumed a width for the engine then they should be able to figure out how fast Manny was driving when he recorded the $0.2 \text{ V}$ measurement: $$v = \frac{\Delta V}{BL} = \frac{0.15 \text{ V}}{5 \text{ mT } \cdot 1 \text{ m}} = 30 \text{ m/s} = 67.1 \text{ mph}$$

The recommendation should be that Manny drive no faster than this. Students should explain to Manny how the potential difference is formed, which is by the Hall Effect.

Discussion Questions

  • Question: How does the right hand rule apply here?
  • Expected Answer: This helps us determine where the mobile charges go. It wouldn't hurt to walk through the RHR again.
  • Question: Draw a free body diagram on a mobile charge in the engine.
  • Expected Answer: Something like the following, depending on whether the charges have fully redistributed yet or not.

Free Body Diagrams

Evaluation Questions

  • Question: How does the problem change if the mobile charges in the statue were positive instead of negative (or vice versa)?
  • Expected Answer: No difference! Instead of negative going to the bottom of the engine, we would have positives going to the top. There is no difference in the charge distribution or how we analyze the problem.

Extension Questions

  • Question: Should gravity factor into the motion of the mobile charges?
  • Expected Answer: It's negligible! Students can show how the magnitudes compare – force of gravity vs magnetic force.
  • Question: Any lingering questions?
  • Question: How are you feeling about the exam?
  • Super Bonus Tutor Question for that Group That Just Can't Get Enough: Model the engine as a thick segment of wire. Make a graph of the current through the segment over time. What is the maximum current through the segment?
  • Expected Answer: Students will have to make some assumptions about the thickness of the segment, and the material from which the engine is made. They may also want to make the assumption that the engine achieves its max velocity instantaneously after being at rest, so that at $t=0$, the engine can be moving at constant speed and the charges will not have redistributed yet. Students will need to use the following equation along with their assumptions: $$I = |q|i = |q|nAuE$$ The only missing part is $E$, the electric field. They should be able to figure out the current drops off exponentially as the charges redistribute, much like it does as a capacitor charges. As far as finding the electric field, this is a little trickier. At $t=0$, the charges have not redistributed, and the only force on a single mobile charge is the magnetic force. This “magnetic force” is actually the same as the electric force due to the relationship between $\vec{E}$ and $\vec{B}$. You don't need to introduce Maxwell's equations or anything, but a cool thing might be to get the students to think of the situation in the constant-velocity reference frame that moves with the engine. Now, the velocity of the engine is $0$, but the charges still move! Since $\vec{v}=0$, the force can't be magnetic, it must be electric. Turns out, they are the same thing! Wowa-weewa. Now, if the initial force on the mobile charge was electric, then the students can find the electric field and then solve for the maximum current.

Alternative Energy Source

The people of Lakeview are starting to get used to living life in the magnetic field, now that it seems to be doing a pretty good job keeping EM-boar tigers from entering the town and causing mayhem. Local alternative energy scientist Alysson Watersa has even installed a source of clean energy that is made possible by the magnetic field. She has constructed a water mill with her bare hands and situated it into the side of the cliff, conveniently below a waterfall so that the motion of the water turns the mill at a frequency of $15 \text{ rpm}$. Dr. Watersa also fixed a board along the axle of the mill, and she has laid a loop of 12-Gauge copper wire (with a radius of $1 \text{ m}$) on the board. One end of the loop becomes a wire that comes out the side of the board and is grounded into the cliff. The other end of the loop becomes a wire that runs out the other side of the mill, all the way back to Lakeview. As the mill turns, so does the loop, causing a current to exist in the wire, which can be used as a small convenience for the good people of Lakeview.

Unfortunately, local superstitious person Manny Callabero is causing a ruckus about the new energy source. He claims that it's all hocus pocus, and Dr. Watersa is just using the water mill as a ploy to hide her stash of jellybeans in the face of the cliff in order to avoid sharing. He won't listen to Dr. Watersa for even a second, so it's up to you to explain to Manny how the water mill works and why it produces the current that it does. It may be helpful for Manny to see a diagram of the mill and a graph of the current produced. Be as detailed as possible, as Manny can be very stubborn about changing his mind. Dr. Watersa's blueprints, shown below, may be helpful.

Diagram

Learning Goals

  • Calculate the induced current from a changing magnetic flux
  • Use the right hand rule to determine the direction of the induced current
  • Explain why induced current is different from the current in a circuit (in other words, what does it mean for a current to be “induced”?)
  • Explain why there is a negative sign in Faraday's Law

This problem is pretty tough, so we basically give students their representation. It shouldn't take them very long to figure out what the motion of the water mill looks like.

The students may need to look up or change the units on a few things:

  • Resistivity of copper is $\rho = 1.72 \cdot 10^{-8} \Omega \text{m}$.
  • Radius of 12-Gauge wire is $1 \text{ mm}$.
  • Alternatively, students may look up that the resistance per length of 12-Gauge copper wire is $5.2 \cdot 10^{-3} \Omega\text{/m}$.
  • The frequency of $15 \text{ rpm}$ rotation is $\omega = \pi/2 \text{ s}^{-1}$.
  • From the previous project, the magnetic field is $B = 5 \text{ mT}$, which is directed into the ground.

Students wish to find current, and it may take them a while to figure out how to do this. If they are stuck, it may be worthwhile directing them to the notes on curly electric field or magnetic flux. The current is induced in the loop because a changing magnetic field produces a curly electric field. An electric field along the loop creates a current. Eventually, they should be able to realize that Faraday's Law is needed, especially the form that includes induced current:

$$I_{ind}R = -\frac{\text{d}\Phi_B}{\text{d}t}$$

At this point, students should know that they need to find flux. We use the following definition, since $\vec{B}$ is constant, and area is flat:

$$\Phi_B = \left| \vec{B} \right| \left| \vec{A}_{\text{loop}} \right| \cos \theta = BA_{\text{loop}}\cos\theta$$

We can visualize theta with the image below. If the board is rotating at a frequency $\omega$, then we could rewrite $\theta = \omega t$ (assuming no phase shift). Students may not realize this at first. Since they must eventually take the time derivative of magnetic flux, they will have to represent something in terms of time. For this problem, that thing is $\theta$.

Theta

So we can write: \begin{align*} \Phi_B &= BA_{\text{loop}}\cos(\omega t) \\ -\frac{\text{d}\Phi_B}{\text{d}t} &= BA_{\text{loop}}\omega\sin(\omega t) \end{align*}

In order to find current, we also need resistance. Students will probability look up resistivity and radius of 12-Gauge copper wire, and use that information to find resistance:

$$R = \frac{\rho L}{A_{\text{cross section}}} = \frac{1.72 \cdot 10^{-8} \Omega \text{m } \cdot 2\pi \cdot 1 \text{ m}}{\pi (1 \cdot 10^{-3} \text{ m})^2} = 0.034 \Omega$$

At this point, students should be able to express the induced current:

$$I_\text{ind} = \frac{1}{R}\left(-\frac{\text{d}\Phi_B}{\text{d}t}\right) = \frac{BA_{\text{loop}}\omega\sin(\omega t)}{R}$$

This is alternating current! We get $I_\text{max} = 0.726 \text{ A}$, and a period of $4$ seconds. A graph is shown below.

Current vs. Time

Discussion Prompts

  • Question: What direction does the induced current flow? For the first half? For the second half? How do you know?
  • Expected Answer: Have students walk you through the right hand rule - see example on separate page. Make sure they understand why the current changes direction - that the flux decreases as the board spins then increases so current changes directions.
  • Question: Why do we need a negative sign in Faraday's Law? (Also called Lenz's Law.)
  • Expected Answer: There needs to be a negative sign in Faraday's Law for energy conservation to still hold. If the negative sign were not there, there would be situations where you get more energy than you started with (aka you can get infinite energy!)
  • Question: What creates this current? What is pushing the charges around the loop?
  • Expected Answer: The electric field produced by the changing magnetic flux creates this current. (It is NOT due to surface charges.) This electric field would point in the same direction as the induced current (curly around the loop and alternating directions with time).
  • Question: What does it mean for the current to be induced? How is it different from the current in a circuit?
  • Expect Answer: When we say induced current we mean the current that is produced by a curly electric field (which comes from a changing magnetic flux). This is different from the current in a circuit because it has a different source. Both currents are produced from electric fields; however in a circuit, the electric field is created by surface charges whereas the induced current comes from the curly electric field.

Evaluation Questions

  • Question: Is the current produced enough to power the town? What could you do to increase the current?
  • Expected Answer: No the current is not very large. We could make B bigger, make A bigger, make it rotate faster, or use more loops.
  • Question: How did you simplify this problem? What assumptions did you make in this problem?
  • Expected Answer: B-field is constant, board rotates uniformly, wire is ohmic,

Extension Questions

  • Question: This is very similar to how real electrical generators work. As an engineering problem, how do you think they prevent the wires from getting twisted up?
  • Expected Answer: Either the ends of the loop are cut and left as conductive brushes which then are free to spin on a conductive plate, or you can rotate the magnetic instead of the loop.
  • Do you have any lingering questions? Anything that doesn't make sense?
  • Expected Answer: TBD
  • 184_projects/s18_project_13_sol.txt
  • Last modified: 2018/04/12 15:53
  • by dmcpadden