2 Flying frogs In lectures, we saw how a magnet could levitate between two bismuth crystals, which are diamagnetic. You can flip things around and levitate diamagnetic objects in magnetic field. Frogs are mostly water which is diamagnetic, so let's take a look at levitating frogs in a Bitter magnet, which is like a solenoid made out of vertically thin, radially wide plates with a hole in the middle that are stacked helically (like a very tight spiral staircase). Feel free to look it up. If you do, you will also see that these plates have additional holes along the edges designed to pour coolants to carry away heat. You may wonder why we're not using 1/3. You'll see later in this problem that using 1/3 will give you an answer that is wrong by orders of magnitude! 1 1. (4 pts) Orient the Bitter magnet so that its axis is aligned with 2 and let B, be its magnetic field pointing upwards along the z-axis. Deduce whether the frog should be placed at the top, in the middle, or below the Bitter magnet in order to float. Hint: what is the force on a magnetic dipole due to an external magnetic field? For this problem, logical deduction and force diagrams suffice to get the answer. 2. (2 pts) Assuming the frog to be a linear magnetic object, what should be its induced magnetic moment? Express your answer in terms of B2, r the radius of the frog (assume spherical frog), Xm the magnetic susceptibility of the frog, and Ho the permeability of free space. 3. (5 pts) Just outside the Bitter magnet (whether above or below), consider h to be the typical length scale over which B2 changes (i.e., dBz/dz ~ Bz/h; in other words, over h, B, changes by order unity). This h should be approximately the radius of the plate. Derive an expression for the required B, to float a frog of density p. Take g as the gravitational acceleration on Earth. Evaluate the magnitude of this field in Tesla using p=2g cm, h= 10 cm, and Xm = -9.0 x 10-6 for water (like humans, frogs are mostly water). (Note: be careful with unit conversion.) 4. (7 pts) This particular Bitter magnet had 1000 plates per metre. Each plate has radius 20 cm with the inner hole radius 2 cm and the entire magnet had a length 30 cm. For the purpose of this exercise, ignore the outer holes used for coolants. Estimate in Watts the amount of power that is dissipated in the Bitter magnet while levitating a frog. Plates are made out of copper so use the resistivity of copper (1/0) ~ 1.68 x 10-8 Ohm m. How many Qubec households does this correspond to? Hint: for the calculation of resistance, try to visualize the path of the current. What is the length of path across each plate (for this exercise, take the larger radius)? What is the length of the path between one plate to another? You can determine all of this using the given parameters. (Note: for this problem, you are expected to provide an answer that is correct within and up to an order of magnitude. If you look up numbers, state the source.) 5. (5 pts) Repeat the exercises 3 and 4 for human levitation (consider an adult human). Enlarge the length and the hole radius of the magnet to ~2 m to "fit" a human. To make the construction of this magnet reasonable, keep the outer radius of the plates to 4 m. Is human levitation by a Bitter magnet practical? (Note: for this problem, you are expected to provide an answer that is correct within and up to an order of magnitude. If you look up numbers, state the source.) Hint: Figure out how much the dimensions of the magnet need to change and scale your answers in 3 and 4 accordingly