Dr. McPaddel woke up to a great surprise this morning when she walked outside to admire the junkyard and noticed a mysterious glowing creature stuck to her magnetic crane. Apparently, in all the excitement of showing the contraption to her neighbors, she had forgotten to turn off the machine. The creature had wandered into the scrapyard overnight looking for some metal to eat, and when it wandered near the crane, the metal became magnetized and pulled the creature up into the air, where it remained stuck until Dr. McPaddel discovered it this morning. She alerted the authorities, but it was the Hawkion researchers who arrived before anyone else, to transport the creature to a research facility to learn more about it. They are thinking these animals might be responsible for the disappearance of fish from the lake and green-bellied canaries from the cliffside.

By the time you learn of all this, the creature is already at the research facility. You arrive and the scientists fill you in. They have acquired this beast to study it, but by now they have figured out Dr. McPaddel's crane snagged a young version of the monster. Because of the apparent resemblance to both boars and tigers and their electromagnetic glow the scientists at the compound have taken to calling the monsters EM-boar tigers. So you have captured an EM-boar tiger cub. It turns out that EM boar tigers are super protective of their young and they have organized to attack the Lakeview compound. Surprisingly, even though it is a government facility, there are no guns at the compound. But you are scientists and engineers godammit! You indicate that maybe capturing this new 'alien' species young might have upset the creatures and make a plan to return the beast but you need a distraction to get the cub close enough to give back to its kin.

You decide to construct makeshift launcher that is using magnetic force to launch projectiles so that they will leave the launcher at a speed of $300m/s$. You must construct your launcher on the floor of the laboratory and position it at an angle of 63 degrees so that it is firing out of the window of the laboratory.

At your disposal to construct your launcher, you have access to two copper rails that can be cut to your specifications. You also have perspex that you can cut to any dimensions and chunks of aluminum that can be cut to any shape. You also have access to multiple variable power supplies that can be set to a requested current and a bunch of wires that can be cut to any length and some switches. You must outline your design, explaining the physics behind the design of the magnetic force launcher and indicating the current needed to reach an exit velocity from the launcher of $300m/s$.

Conceptual questions:

1. In this scenario, you used the F = integral I dL x B equation. In particular, what current/lengths did you pick? What is making the B-field? The force that you found is the force from what on what?
2. There are a couple key assumptions you need to make about the B-field in this problem. What assumptions did you need? Why did you need them? How realistic are those assumptions?
3. What sort of right hand rules did you use in this problem? Why did you need two different ones?
4. What steps did you take to solve this problem?
5. What size of current did you end up finding for the problem? Is it big/small/normal? How would that impact the practical use of railguns?
6. Make a summary of the equations that we've talked about in class so far (field & force for point charges & currents). What is similar about the equations? What is different about the equations? When are each of the equations applicable?

Despite your best efforts to return the cub, the EM-Boar tigers around Lakeview have started causing all sorts of problems. Last night, one of them came through Lakeview and tore down all the power lines, including the transformer for the incoming transmission line from the power plant on the edge of town. All the wires are chewed up and torn, and there are deep claw marks on the utility poles, which luckily are still standing. The residents of Lakeview are pretty spooked. Lakeview needs its power back as soon as possible.

There is an incoming transmission line on the edge of town from which you need to set up the power lines that will run through all of Lakeview. The manager of the power plant, Dr. Erma Cürd, has supplied you with some specifications: Each power line will be erected to connect the transformer from the incoming transmission line to the homes and businesses in Lakeview. Each line is $5 \text{ km}$ long and is made of a metal alloy with $0.008 \text{ $\Omega$/m}$ of resistance.

The most important decision in this reconstruction process is to determine which transformer to install at the incoming transmission line. A given transformer will create a specified voltage drop from the transmission to the residential area. However, there are some risks associated with your choice. One risk is that the electric field along the line will heat up up the wire and cause it to melt, which will happen when the electric field reaches $3 \text{ kV/m}$. Another risk is that the line may create a magnetic field on the ground that is dangerous for people walking around and may interfere with portable electronics. The safety limit for the magnetic field is $10 \text{ mT}$.

You have three options for your choice of transformer. The voltage drop on the line from the functioning transmission line to the residential area can be $1 \text{ MV}$, $10 \text{ MV}$, or $100 \text{ MV}$. Evaluate each decision and produce a recommendation based on the safety concerns and the power that the transformer will produce for Lakeview.

Conceptual questions:

1. In your calculations, you used dl or L a couple of times. What equations did you use with lengths, and which lengths were they referring to?
2. What steps did you need to take to simplify the $\int \vec{B} \bullet d\vec{l} $?
3. How did you pick your Amperian loop? Would a square Amperian have worked for the power line? Would a circle that is off center work?
4. How would you calculate the magnetic field inside the wire? What would change about your calculation?
5. If you had two power lines side by side, how would you find the force from one wire on another?
6. When do you want to use Ampere's Law and when would you want to use the Biot-Savart Law to find magnetic field?