184_projects:f20_project_4_sol

warninglight.jpeg

You are working for The Jet Propulsion Laboratory (JPL) Division of NASA testing a new and highly experimental spacecraft capable of in atmosphere flight as well as outer orbit maneuvering. Lieutenant Pete “Maverick” Mitchell has been testing the new spacecraft now for a few weeks, and has had continuous issues with the warning light for power delivery failure to the stabilizers on the wings.

Maverick is convinced that the time between when the problem happens and when the warning light comes on is way too long, and this delay in relaying the warning to him could lead to a very problematic incident in the future. He does not want to loose another Goose. You have been given the task of explaining to Maverick why this could not be the case, and your boss, Clint Howard, has given you the following circuit diagram to try and aid you in your explanation to Maverick. The circuit diagram at this time does not include the warning light.

Maverick likes numbers, so part of your explanation should also include a calculation of the amount of time it takes the light to come on when the length of the wire in the circuit between the stabilizer control module switch and the warning light in the cockpit is 5.2$m$ distance. He should also understand what the electric field looks like in this circuit, and why it means that his original presumptions about the warning light “taking too long” cannot be correct. You should also correct this circuit diagram to include the warning light so that your boss has a more accurate circuit diagram to show people.

Learning Goals

  • Explain what happens to the surface charges and electric field in a circuit when wires are initially connected.
  • Explain why current starts to flow almost instantaneously (or rather why a light bulb turns on immediately after you flip the switch)
  • Explain why a light bulb would not turn on if it were only connected to the positive and negative of the battery.
  • Explain the role of the battery in lighting up the light bulb.

Learning Issues

  • Most of the learning goals this week are “explain” based learning goals. This problem is much more conceptually focused than previous problems. It's probably worth warning your students at the beginning of class that the goal of this problem is different.
  • If students are struggling with where to start ask them to first draw what the surface charges look like on the wires (using the fact that the wires are conductors).
  • $\Delta$t in the notes is NOT the time to run on the light bulb - this is the average time between collisions of the electrons in the wire.
  • Students may struggle with the distinction between surface charges and the charges moving in the wire (current) - these are different things!
  • It is easy to glaze over the intermediate steps in this problem. Make sure that students are explaining how the gradient of surface charges is set up, not that it just happens.

Timing Issues

  • Watch how much time you spend with each group. It's easy to spend 45 minutes with one group and ignore the other completely with this problem due to the conceptual focus. Students are going to ask more questions, which is fine, but be aware of the balance.

Other Issues

  • This is the day after the exams - students are probably worn out and may not be as prepped for this week after focusing on studying for the exam. Be aware of this.

This is a very qualitative solution, which is different than most of the previous solution. Make sure you have the groups thoroughly explain what is happening and why, and also draw pictures for the surface charge distributions.

Before the wires are connected

Suppose we have a battery, with one positive end and one negative end (could be either the chemical model or mechanical model of the battery). At each end, we connect a conductive wire. Before we connect the wires, what would the charge distribution look like along the wire?

The wire that is connected to the negative end of the battery would have a negative distribution of charge because the excess electrons would be able to spread along the length of the wire. Since the wire is a conductor, those excess charges would spread out along the surface of the wire, leaving the interior of the wire with a net electric field equal to zero. A similar process would occur for the wire connected to the positive end of the battery. An approximate charge distribution is shown in the figure, ignoring the exact details of the charge distribution.

If we zoom in on the gap between the wires, We can see an electric field at the end of the wires; however that electric field is canceled out by the electric field from the wire across the gap. So even in the ends of the wires, the net electric field is zero. There is a non-zero electric field only between the ends of the two wires.

  • Question: In the broken circuit diagram that you are given at the beginning of the problem can you sketch the electric field? What does the electric field look like between the two ends of the wires?
  • Expected Answer: The electric field points directly from the positive end of the wire to the negative end of the wire. (see above)
  • Question: At this point is there current flowing in the wire?
  • Answer: No. The electric field is zero in the wire. (There may be a small/short current when the charges from the plates spread out over the wires, but not after this process is over.)

Immediately after the wires are connected

If we complete the circuit by connecting the wires, what happens to the surface charges in the wire? The charges on the ends of the wires neutralize each other (so there is no net charge in the middle of the wires). This leaves a “tube” of surface charge on the outside of the wire. In the instant that the two wires touch, there is a strong discontinuity in the charge distributions along the wires - one side of the wire is positive and one side is negative with no smooth transition between the two. This means that there is a separation of charge and therefore there is now an electric field in the now-touching ends of the wires. The electric field in the rest of the wire is still 0. The electric field where the wires meet will point from the positive charges to the negative charges. Close to the edges, the electric field will point slightly toward the negative charge (like with the line of charge). This electric field will pull some of the negative surface charge toward the positive surface. This will create a small neutral zone which becomes important in the next time snapshot.

  • Question: What does the electric field look like immediately after the light bulb has been inserted and the connection has been made?
  • Expected Answer: The electric field points toward the center of the wire on the positive side and toward the edges on the negative side.
  • Question: What creates the constant electric field inside the wires?
  • Expected Answer: The surface charges create the electric field. There is a constant change (gradient) in surface charges along the wire to create the constant electric field.

A moment after the wires are connected

This neutral zone makes a couple of things happen. First, the electric field where the wires meet expands because the area between the positive charges, the neutral zone, and the negative charges is larger than when the charges have not cancelled out. This increased area between the regions causes the surface charges to spread out some because the positive charges will be repelled by the other positive charges further up the wire and move towards the negative region. The same is true about the negative charges. There are now parts of the positive wires which have more positive charges than the part close to where the wires meet. The electric field “spreads” because of this reason: the forming gradient causes there to an electric field further up the wire that points from more positive regions to less positive regions.

Because there is now an electric field in the wire, the mobile sea of electrons will start to move, which changes the surface charge distributions. This is a result of the neutral zone of surface charge discussed in the previous instant. Note that this starts happening after only fractions of a nano-second. Electrons moving toward the positive region of the wire will make the surface charge in that region less positive. Likewise, electrons moving away from the negative region of the wire will make that region less negative. In the middle of the wires, this creates a region with smaller amounts of charge, making that discontinuity of positive and negative charges less drastic, and setting up an electric field between the positive and negative regions.

A short time after the wires are connected

Again moments later (fractions of nano-seconds), the charge distributions in each side of the wire have rearranged. The excess negative charges have spread out so they have shifted away from the battery and more towards the center of the wire, with a similar process occurring for the positive charges. This creates regions in the wires where close to the battery there is a more concentrated surface charge and closer to the center there is a weaker surface charge. This sets up the continuous charge distribution that we see for the steady state. With this continuous charge distribution, there is now an electric field set up within the wire, which starts to move the electrons in the electron current.

The surface charges spread out some because the positive wire will repel some of the positive surface charges near the end (like charges repel, so the concentration of positive charges in that wire will spread some). The same thing happens with the negative wire. This spreading of charges bumps up against the neutral zone where there is less positive charge present. This means that there is now part of the positive wire that has more positives than the part close to where the wires meet. So, and electric field is present further up the wire that points from the more positive region of the wire to the less positive region of the wire (establishing a small, local gradient of charge). The same process occurs in the negative wire simultaneously. This electric field then starts to push the electrons in the middle of the wire and move the surface charges.

How long does this process take?

The electric field from the surface charges is set up between the two wires almost instantaneously. That electric field immediately starts pushing on the electrons in the region (from the electron sea), which then move. The electrons do not have to move very far to change the surface distribution of charge, which extends the electric field around the circuit. This causes a push on the electron in the region of the extended electric field, and so on. This means that the electric field is able to travel around the wire much more quickly than the individual electrons in the wires. In fact, the electric field travels at the speed of light ($3*10^8$ m/s) students may not know this ahead of time and may need prompting to look up, while the electrons in the wires are only traveling at close to $5*10^{-5} m/s$. So the circuit reaches a steady state long before a single electron makes a whole loop around the circuit. Once the steady state scenario has been reached, the electric field inside of the wire is constant. At that point, the only charges which are moving are the charges moving through the bulk of the wire. The surface charges at this point in time are stationary. The constant electric field is both constant in magnitude and direction (with the exception of the corners, or the kinks, in the wires, but we can make an assumption to ignore this). The current in the wire is constant because the electric field is constant and there is a constant force on the charges in the bulk of the wire (relatively speaking).

This explains why when you turn on a switch, the light comes on almost immediately. You are not waiting for an electron from the switch to travel all the way to the light bulb. You are waiting for an electric field to be set up in the wire to push the electrons that are already in and near the light bulb. Thankfully that process is much shorter! Thus, it is the speed that the electric field moves through the wire that determines how quickly the current starts, not the speed of the electrons themselves.

Discussion Prompts

  • Question: Where does the current start flowing?
  • Expected Answer: Electric field starts in the middle where the wires are connected, thus the charges first start moving in the middle of wire. But the electric field spreads almost immediately through the wire, so the charges everywhere in the wire move almost immediately. The charges do not start from the battery, but the electric field in the wire pushes the electrons that are already in the wire.
  • Question: Based on what you found, what does it mean for the circuit to be in a steady state?
  • Expected Answer: Steady state happens when the surface charges are no longer moving. This means that the electric field in the wires is CONSTANT and makes a constant current everywhere in the circuit. It is automatically not a steady state if the current is changing.
  • Question: How fast does the electric field travel compared to the drift velocity of the electrons?
  • Expected Answer: The drift velocity of the electrons is much slower. The drift speed is approximately $10^{-4}-10^{-5} m/s$ whereas the electric field travels at the speed of light.
  • Question: Why does the light bulb turn on quickly when the switch is flipped if the electron velocity is so slow?
  • Expected Answer: Because the field travels fast, which starts pushing electrons in all parts of the wire almost instantly. So a current is established everywhere within nanoseconds, even though it may take a very long time for an electron to start at the battery and reach the light bulb.
  • Question: Why does the light bulb turn on quickly when the switch is flipped if the electron velocity is so slow?
  • Expected Answer: Because the field travels fast, which starts pushing electrons in all parts of the wire almost instantly. So a current is established everywhere within nanoseconds, even though it may take a very long time for an electron to start at the battery and reach the light bulb.
  • Question: What is the role of the battery in lighting up the light bulb? (It may be easier to answer this question as: what would happen if that battery were removed?)
  • Answer: Initially the battery provides the excess surface charges, which then creates the electric field in the wire. Afterwards, the battery maintains the surface charge gradient and provides the electrons that move around the wire (i.e. the electrons in the electron current.)

Evaluation Questions

  • Question: In doing this problem, we have made a model of how current flows in a circuit. What are some of the things we have simplified/ignored in this situation? (Assumptions and approximations?)
  • Answer: We did not talk about how the electrons bend around the corners of the wires, we simplified the battery to a pair of charged parallel plates (keeping the voltage constant), we're simplifying a cylinder of surface charges to a couple +/-'s on each wire (this really happens in 3D around the surface of the wire).

Extension Questions

  • Question: Would a light bulb turn on if it were only connected to the positive end of the battery?
  • Answer: No. If this were true there would only be positive surface charges, which means the electric field inside the wire would be zero - therefore, there is nothing to push the electrons through the wire so there is no current and the light bulb would not turn on.
  • Question: Any lingering questions for this week? Anything still confusing?
Testing Observations

We (meaning Daryl) had too much about this process written into the notes. Since then, this page of notes has been removed and students are expected to reason through this on their own.

  • 184_projects/f20_project_4_sol.txt
  • Last modified: 2020/09/29 19:05
  • by dmcpadden