Extra Problem A5 for 7D for Chap 28 (and chap 27), corresponding to MP A6 [just one this week]. Please upload your solution to the course website by 11:59 Tuesday 5/23/2023 In this problem we consider continuous currents, rather than current in individual wires. Suppose we have an insulating hollow cylinder of radius R, and the central axis of the cylinder is up on the page (which we define as the z direction). A total charge of Q is uniformly placed on the cylinder. Now suppose the cylinder rotates with an angular velocity W1= W1 Z. The spinning charge creates a current that flows around the cylinder, much like a wire in a solenoid. Assume the length of the cylinder is L, which is long enough to use Amperes Law to find the strength of the B field inside the cylinder (so L>>R) A) How do you compute the current? The definition of current is the total charge that flows past a point in a wire per unit time. In this case, you can imagine a vertical line on the surface of the cylinder, and compute the time it takes to complete one revolution. In this time, you know that all of the Q on the cylinder has flowed past this point. So I=Q/(time to complete a full rotation). What is the total current? B) Review the calculation of the B field in a solenoid, where B=onl. For that calculation, n is the number of wires per unit length, and I is the current in each wire. For the present calculation, you do not know I, the current in each wire. But notice the combination of quantities, nl, is current per unit length. What is the current per unit length for this cylinder? From this information, what is the magnitude of the B field inside the cylinder? What is the direction? C) Next, consider a thin insulating disk of mass M and radius R2, with charge Q2 uniformly placed on the disk. The disk lies flat on the table. The disk rotates long an axis that is normal to the disk, and intersects with the center of the disk. It spins at angular velocity W2. The spinning creates a continuous current. What is the magnetic moment, , of this disk? Recall that for a loop of wire carrying a current, the magnetic moment is I*A. But for this disk, the current is continuous. Like with continuous charge distribution, one way to determine the magnetic moment is to break of the current into chunks, and it should make sense that the chunks are thin circular rings. So the key is to determine the current at an arbitrary radius r. Then multiply the chunk of current in the thin ring by the area within that radius r to get a chunk of the total magnetic moment. Then integrate the chunk of magnetic moment from 0 to R2. To do this plan, determine the charge contained in an thin ring of area of dA= 2rdr, where r is the radius of the ring. Like in parts A) and B) above, you know all the charge in the ring will flow past a point in this annular "wire" in the time to complete one revolution. This allows you to calculate the chunk of current dl at radius r (where I symbol is current). Do NOT compute the B field from the disk, only the magnetic moment... D) Finally, suppose you place the center of the rotating disk inside the rotating cylinder along its central axis (to be able to do this, R2 < R). The normal to the disk is tilted at an angle, 0, relative to the direction of the B field in the x-z plane. What is the magnitude and direction of the torque? Just for thought. What do you think is the motion of the rotating disk inside the rotating cylinder? In part D you should have found a torque on it, which presumably causes a rotation. But the rotation direction of the spinning disk is very surprising. Take a look at section 10.7 in your textbook for a hint. The key idea is that a spinning disk has angular momentum, L, and it can only change in the direction of torque vector t=dL/dt)