184_projects:s18_project_14_sol

rainbow.jpg

Life has gotten back to normal in Lakeview as the continued magnetic field seems to have driven away the boar tigers. There has been the added bonus that there has been a drastic reduction in car accidents due to the populace being afraid that the magnetic field will destroy their engines. However, tranquility is unsettled when the government agency P.E.R.L. comes to town in order to further investigate the boar tiger phenomenon. You are a local guide, part-time engineer and a high ranking member of the McGyver fan club who has been recruited to go into the forest surrounding Lakeview to help P.E.R.L in their investigation. Once deep in the forest, your party is ambushed by boar tigers and several of the team are captured and taken away. The remaining members of the team try to follow the tracks of the Boar Tigers but this becomes impossible after a time. However, you remember hearing from a friend of yours that did research on the boar tigers that their excrement has a very high percentage of nickel in it. You indicate to the group that you think you can McGyver a metal detector together to track your kidnapped colleagues. You have access to a whole bunch of 12 gauge copper wire, an alternating current power supply that can produce a maximum current of 2 A and has a frequency of 60 Hz, and a voltmeter that will measure an alternating voltage. If the voltmeter measures a voltage bigger than 50 mV, it will sound an alarm. You want to design your metal detector so that if there is not a metal nearby, the voltmeter will be below the threshold; however, if there is a metal nearby, then the measured voltage on the voltmeter will be above the threshold and set off the alarm.

There are several ways to do this problem. One of which (described below) uses two concentric coils (just like the first page of notes). You would connect the alternating current to the larger outer coil and connect the voltmeter to the smaller inner coil. Since the current is constantly alternating, the large coil induces a current/voltage in the smaller coil. When close to metal the magnetic field from the metal will increase the flux through the small coil and increase the induced voltage.

For this case, we would write the current in the large coil as: $$I_{lg}=I_0sin(2\pi \cdot f \cdot t)$$

We then use that current to find what the magnetic field would be at the center of the large coil (with $N_{lg}$ loops): $$B_{lg}=\frac{\mu_0 I_{lg}N_{lg}}{2 R_{lg}} \hat{z}$$ $$B_{lg}=\frac{\mu_0N_{lg}I_0}{2 R_{lg}}sin(2\pi \cdot f \cdot t) \hat{z}$$

This means that the magnetic field is increasing and decreasing over time and switching between positive and negative values. Based on what we know from Faraday's Law, this changing magnetic field will create a curly electric field. Since we are looking at the changing magnetic field in the center of the large loop, we are also talking about the curly electric field at the center of the large loop. If we place a smaller coil (with $N_{sm}$ coils) in the center of the large coil, the changing magnetic field (from the large coil) will induce a current in the smaller coil.

$$-V_{ind}=\frac{d\Phi_{B}}{dt}$$ where here we are looking at the $V_{ind}$ in the small coil which means we are looking at the changing magnetic flux through the small coil. If we separate the flux so it's in terms of the magnetic field and area, then the flux we are talking about is due to the magnetic field from the large coil ($B_{lg}$) that passes through each of the small coils ($N_{sm}\cdot A_{sm}$), so we have: $$-V_{ind_{sm}}=\frac{d}{dt}\left({\int \vec{B}_{lg}\bullet d\vec{A}_{sm}(N_{sm})}\right)$$ We have already said that the magnetic field points into or out of the page, which is parallel to the $d\vec{A}$ of the small loop (remember the area vector would need to be perpendicular to the small coil). This means our dot product simplifies to simply multiplication:

$$-V_{ind_{sm}}=\frac{d}{dt}\left({\int B_{lg}(N_{sm}) dA_{sm}}\right)$$

Now, we will assume that the magnetic field in the small coil is approximately constant through the area (which is roughly true if the small coil is much smaller than the large coil). This means that we can pull the $B_{lg}$ out of the integral - but we can't pull it out of the derivative because this magnetic field is not constant with time! This makes the integral part of the flux very easy since $N_{sm}$ is also a constant.

$$-V_{ind_{sm}}=\frac{d}{dt}\left(B_{lg}{\int (N_{sm}) dA_{sm}}\right)$$ $$-V_{ind_{sm}}=\frac{d}{dt}\left(B_{lg}N_{sm}A_{sm}\right)$$

Now we have to think about this derivative with respect to time. The only thing that is changing with respect to time is the magnetic field from the large coil, so we can pull the area and number of turns out of the derivative.

$$-V_{ind_{sm}}=N_{sm}A_{sm}\frac{dB_{lg}}{dt}$$

Now we can plug in the magnetic field for the large coil that we found above, and pull any constants out of the derivative: $$-V_{ind_{sm}}=N_{sm}A_{sm}\frac{d}{dt}\left( \frac{\mu_0N_{lg}I_0}{2 R_{lg}}sin(2\pi f \cdot t)\right)$$ $$-V_{ind_{sm}}=N_{sm}A_{sm}\frac{\mu_0 N_{lg} I_0}{2 R_{lg}} \frac{d}{dt} \biggl(sin(2\pi f \cdot t)\biggr) $$

This derivative is then fairly simple to take: $$\frac{d}{dt} \biggl( sin(2\pi f \cdot t) \biggr) =2\pi f cos(2\pi f \cdot t)$$

So we get the induced potential in the small coil to be: $$-V_{ind_{sm}}=N_{sm}A_{sm}\frac{\mu_0 N_{lg} I_0 2\pi f}{2 R_{lg}}cos(2\pi f \cdot t)$$

We want to set this voltage to be very close to the 50 mV threshold so that if a piece of metal is nearby the induced voltage will increase and cross the threshold. Students can adjust $N_{sm}$, $A_{sm}$, $R_{lg}$ and $N_{lg}$ to get the maximum $V_{ind}$ to be close to the threshold.

Discussion Questions

  • Question: How did you create your meatl detector?
  • Expected Answer: There should be some sort of coil, loop or solenoid that has the oscillating current and another sort of coil or loop that has an induced voltage. When brought near the metal there should be a change (increase) in the induced voltage compared to when the metal is not there.
  • Question: When the current is increasing in the outer coil, what direction is the induced current in the inner coil? (Depends on their set up)
  • Expected Answer: Walk through the right hand rule for induced current with them. (The answer is highly dependent on the set up they come up with.)
  • Question: Make a graph of the current in the outer coil compared to the induced current when there is not metal nearby.
  • Expected Answer: See instruction sheet - one should be sine curve and the other should be a negative cosine curve (or vice versa).
  • Question: Does the current in the outer coil and the current in the inner coil always point in opposite directions?
  • Expected Answer: NO! The induced current is related to the change in magnetic flux (NOT directly to the magnetic field or the current)

Evaluation Questions

  • Question: What would you expect to change about your graph if a metal is placed nearby?
  • Expected Answer: Only the amplitude of the induced current should change (or get larger). The frequency should not change. Nothing about the current in the larger coil should change (since it's hooked up to the power supply).
  • Question: How did you pick the numbers for your metal detector design?
  • Expected Answer: We know the area of the large loop should be much bigger than the area of the small loop because we assumed the magnetic field is constant through the area of the small loop. Then we picked the number of turns so that $V_{ind}$ was close to the threshold.

Extension Questions

  • Question: What would happen if you had a different kind of material?
  • Expected Answer: Have students look up the different kinds of magnetic materials (ferromagnetic, paramagnetic and diamagnetic).
    • A lot of metals are ferromagnetic - so the magnetic domains strongly align with the magnetic field they are in. These would cause an increase in the induced voltage in the metal detector.
    • Some metals (and other materials) are paramagnetic - domains weakly align with the magnetic field; the metal detector may not be able to detect these (it would have to be very sensitive to changes in the field).
    • Many materials (and some metals) are diamagnetic - which means the domains align against the magnetic field. This would cause a decrease in the induced voltage. However, this is usually a very small effect and may not be visible. Most materials that you would think of as “not magnetic” are classified as diamagnetic (i.e. wood, plastic).
  • Question: Any lingering questions about Faraday's Law?

You are able to use your successfully MacGyver'd metal detector to track what the EM-boar tigers have left behind and find your trapped team members. You estimate there is about 1 hour and 50 minutes to figure out a way to send some sort of E&M signal out for rescue before the EM-boar tigers realize that you are planning an escape. (What will that signal be??? You decide.) Due to the sensitivity that EM-boar tigers have to E&M waves, you only have one chance to send the signal before they realize what is happening and close in on the team. You will need to make sure that the attempt will work. You have a small metal ball that can be charged up 1e-6 C without discharging in air. You've found a piston that can shake the ball a total distance of 2cm, but at nearly any frequency you need. Your team decides to create a model of the signal to determine how best to orient the setup, how much charge to dump on the ball, and how quickly to shake it. Your group has already started setting up the code.

GlowScript 2.6 VPython

## Objects

charge = sphere(pos = vector(0,0,0), radius = 0.001)
charge.v = vector(0,0,0)

## Constants and model parameters

q = 1e-6
k = 9e9
mu = 4*pi*1e-7
mofpi = mu/(4*pi)

## Set up time parameters

t = 0
dt = 0.1


## Create list of arrows that encricle the charge
## Each location has two arrows: one for E and one for B

N = 20
theta = 0
dtheta = 2*pi/N
R = 0.02
ArrowList = []

while theta < 2*pi:
    
    Loc = vector(R*cos(theta), R*sin(theta), 0)
    
    ArrowList.append([arrow(pos=Loc, axis = vector(0,0,0), color=color.cyan),arrow(pos=Loc, axis = vector(0,0,0), color=color.magenta)])
    
    theta += dtheta

## Calculation loop

while t < 1000:
    
    rate(100)
    
    ## Charge should oscillate
    
    
    ## Loop through arrows to make E and B vectors
    
    for Arrow in ArrowList:
        
        r = Arrow[0].pos - charge.pos

        E = vector(0,0,0)
        B = vector(0,0,0)

        Arrow[0].axis = E
        Arrow[1].axis = B
       
    t = t + dt

In this project, students have to do three things:

  1. Model the motion of an oscillating charge in 1D
  2. Calculate the electric and magnetic fields associated with that charge
  3. Scale the electric and magnetic fields to be able to see them as arrows

The minimally working program sets up the scene, creates the charge and the list of arrows that will represent the electric and magnetic field, and has the nested loop structure for making the charge oscillate and updating the arrows.

This a very good project to have students make predictions about what the electric and magnetic field field will look like at a single point as the charge oscillates. This will help them know if their code is working properly.

Making the charge move

To make the charge oscillate, students will need to model it's motion sinusoidally: $y(t) = A \sin \omega t$. They are free to choose $A$ and $\omega$, but the problem structure suggests $A = 1$cm. The choice of $\omega$ is tricker because it can be anything and if it's too high, the $dt$ value needs to be lowered. So, suggest to them something that is a ten or a hundred $dt's$.

The resulting code appears in the first part of the loop.

charge.pos.y = A*sin(omega*t)
charge.v.y = omega*A*cos(omega*t)

Notice that they need the y-velocity as well because they will need that for the magnetic field. Also, they must define $A$, $omega$ somewhere.

Depending on when students took Physics 1/Calc 1 they may not remember the the derivative relationship between velocity and position. You may have to remind them of this.

Modeling the electric and magnetic fields

This is a process that they should be comfortable with at this point: modeling the fields due to point charges. They might want to include the fact that the charge is moving, but that's not needed as the regular point charge formula will work. These need to be done for every arrow, so they will appear in the nested loop structure. We've tried to indicate that by calculating $\mathbf{r}$ there in the minimal working program. So they need to add,

E = k*q*r/mag(r)**3
B = l*q*cross(charge.v,r)/mag(r)**3

Technically, these models are incorrect as relativity tells us that the field at a location $r$ will be the field produced by the charge at a time $t-r/c$ where $c$ is the speed of light. This is the retarded field as the field travels at the speed of light and the field produced at $r$ is not instantaneously updated. This is a interesting point of discussion, but not a big deal for the scale of the problem we have as $r$ is small.

Scaling

This problem is challenging to visualize because the scale of the $E$ field and the $B$ field are so different (in spatial terms). The students will have to scale both fields in different amounts to see them. The $E$ field will have to be scaled down and the $B$ field will have to be scaled up. For a choice of $\omega = 1$, scales for $E$ and $B$ were 1e-9 and 1e10 respectively!

Evaluation Questions

  • Question: What direction would you expect the electric field to point? What direction would you expect the magnetic field to point?
  • Expected Answer: E-field should point away from positive charges. B-field should point around the charge (if it's moving in the z direction). They should be perpendicular to each other.
  • Question: What would you expect to change if the charge oscillated up and down instead into and out of the page.
  • Expected Answer: B-field would change direction (into and out of the page by the right hand rule), E-field magnitude would change depending on separation, but the direction would stay the same.
  • Question: What would you expect to happen if you oscillated faster? How would that change the EM wave?
  • Expected Answer: If it oscillates faster it will have a higher frequency, which would correspond to an EM wave with higher energy (i.e., gamma or x-ray)
  • Question: What would you expect to happen if the oscillation distance is longer (i.e. meters instead of cm)? How would that change the EM wave?
  • Expected Answer: This would change the amplitude of the signal, making the wave stronger (but would not change the frequency).

Discussion Prompts

  • Question: What happens to these fields far away? What could you do to make sure the fields are large enough when you are far away?
  • Expected Answer: Eventually the fields would die off (because of damping that happens with the air particles). This should let you talk about radio antennas, which are basically long straight rods where the electrons are forced to oscillate up and down the rod. Radio antennas have long travel lines (10-100m) for a strong signal and (relatively) low frequency (kHZ to MHz). This lets the signal travel long distances without dying off.
  • Question: We've assumed that the electric field changes automatically. Does this happen in real life? Hint: if you are really far away from the oscillating charge, would you see the fields change immediately?
  • Expected Answer: No, the fields would take some time to reach you. It turns out that fields travel at the speed of light, so while you would see the change quickly (depending on how far away you are), it would not be instantaneous.

Extension Questions

  • Question:What is FM? AM? What happens to the signal in these cases?
  • Expected Answer: Ask them to look up modulation, what does it do.
  • Question: Any lingering questions?
GlowScript 2.6 VPython

## Objects

charge = sphere(pos = vector(0,0,0), radius = 0.001)
charge.v = vector(0,0,0)

## Constants and model parameters

q = 1e-6
k = 9e9
mu = 4*pi*1e-7
l = mu/(4*pi)

A = 0.01
omega = 1
Bscale = 1e10
Escale = 1e-9

## Set up time parameters

t = 0
dt = 0.01


## Create list of arrows that encricle the charge
## Each location has two arrows: one for E and one for B

N = 20
theta = 0
dtheta = 2*pi/N
R = 0.02
ArrowList = []

while theta < 2*pi:
    
    Loc = vector(R*cos(theta), R*sin(theta), 0)
    
    ArrowList.append([arrow(pos=Loc, axis = vector(0,0,0), color=color.cyan),arrow(pos=Loc, axis = vector(0,0,0), color=color.magenta)])
    
    theta += dtheta
    

## Calculation loop

while t < 1000:
    
    rate(100)
    
    ## Charge should oscillate
    
    charge.pos.y = A*sin(omega*t)
    charge.v.y = omega*A*cos(omega*t)
    
    ## Loop through arrows to make E and B vectors
    
    for Arrow in ArrowList:
        
        r = Arrow[0].pos - charge.pos

        E = k*q*r/mag(r)**3
        B = l*q*cross(charge.v,r)/mag(r)**3

        Arrow[0].axis = Escale*E
        Arrow[1].axis = Bscale*B
       
    t = t + dt
  • 184_projects/s18_project_14_sol.txt
  • Last modified: 2018/04/19 14:36
  • by dmcpadden