Name: ________________________________________________________ 1. Ampere’s Law (10 points). A solid cylinder with radius a is coaxial with a tube with inner radius b and outer radius c. The central cylinder and tube carry equal currents I in opposite directions. The currents are distributed uniformly over the cross sections of each object. Derive an expression for the magnitude of the magnetic field (a) inside the central cylinder (i.e., r < a) (b) outside the central cylinder but inside the tube (i.e., a < r < b) (c) within the body of the tube (i.e., b < r c). SHOW YOUR ORGANIZED AND CLEARLY EXPLAINED WORK!!! AGAIN, YOUR GOAL ISN’T TO WRITE DOWN THE RIGHT ANSWER…YOUR GOAL IS TO CONVINCE ME THAT YOU UNDERSTAND THE MATERIAL AND THAT YOU KNOW WHAT YOU’RE DOING. Physics 200, E&M, Spring 2017 Name: ________________________________________________________ 2. Inductance and Biot-Savart (10 points). A small circular metal ring of radius r is concentric with a large circular metal ring of radius 10r. Current in the outer ring flows counterclockwise due to an unpictured power supply. By adjusting the power supply, you can adjust I, the current in the large ring. The graph below shows I(t). Notice that I increases linearly from I0 to 2I0 from time t = 0 to time t = T. (a). On the same graph, sketch a qualitative graph of the magnitude of the current in the small ring as a function of time. You don’t have to get the magnitude on the current axis exactly right, but you do need to get the overall shape, the zero level, and the timing correct. (b). On the circular diagram, indicate the direction of the induced current flowing in the smaller ring. In a sentence or two, justify your reasoning. An arrow indicating the direction but without any reasoning will get no credit. (c). Using the Biot-Savart rule, find the magnetic field B (magnitude and direction) due to the current in the large ring at the center of the small ring when the current in the large ring is 1.5 I0. The direction can be described by words such as “into the page”, “out of the page”, “in the plane of the page”, etc. (d). When the current in the large ring is 1.5 I0, what is the current in the small ring? Answer in terms of R (the total resistance of the small ring), r, I0, T, and any relevant universal constants. You can regard the B field can be taken as uniform over the area of the small ring, and you can neglect the small ring’s self-inductance. Physics 200, E&M, Spring 2017 Name: ________________________________________________________ 3. Magnetic Fields (10 points). Two current carrying wires are shown below. A current I1 is directed into the page, as shown. A current I2 is bent into an arc, as shown. (a) What is the magnitude and direction of the magnetic field at the point P? (b) Imagine a third current I3 passing through a wire coming out of the page at point P and parallel to I1. What is the force or force per unit length on the wire carrying I3 in the plane of the page (i.e., at z =0)? Physics 200, E&M, Spring 2017 Name: ________________________________________________________ 4. Magnetic Moment (10 points). Consider a non-conducting disc or radius R and uniform surface charge density . The disc rotates with an angular velocity . (a). Calculate the magnetic moment vector μ in terms of . (b). Recall from all the way back to last semester that τ = Iα and that α = dω/dt. If the mass of the disc is M0, what is the magnitude of the constant torque necessary to bring this disc to a stop in a time T? (c). What is the external magnetic field Bext, if any, that can bring about the torque from part (b)? Explain your reasoning…an answer without an explanation will get zero credit. Physics 200, E&M, Spring 2017 Name: ________________________________________________________ 5. Faraday’s Law (10 points). A loop of wire with a 4.00 cm diameter and a 0.500 Ω resistance surrounds a solenoid. The solenoid has a diameter of 2.00 cm, is 10.0 cm long and has 150 turns. The current in the solenoid increases linearly from 100 A to 30.0 A in 3.00 seconds. What is the direction and magnitude of the current induced in the loop of wire? Physics 200, E&M, Spring 2017 Name: ________________________________________________________ 6. Motional EMF (10 points). Consider a straight metal rod of length 2l that is rotating about its midpoint in a plane that is perpendicular to a uniform magnetic field, B, as shown in the left figure. The rod is rotating with an angular velocity of ω. (a) What is the induced emf ε between the midpoint of the rod and each end of the rod. What is the induced emf ε between the two ends? (b) If we hook up this rotating rod as shown on the right, with an electrode connected to the middle and another electrode connected to a ring track that maintains constant electrical contact with both ends of the rod, what is the current passing through the resistor R. (c) Instead of a rotating rod, consider a solid rotating metal disc of radius l and angular frequency  rotating in a uniform magnetic field B. If we hook up this disc as in (b), what is the current through the resistor? This type of current source is called a homopolar generator. R . Physics 200, E&M, Spring 2017 Name: ________________________________________________________ 7. Biot-Savart Law (10 points). Find the magnetic field B for all points on the axis (i.e., an arbitrary distance z from the plane of the loop) of a circular loop of wire of radius R and carrying current I. Physics 200, E&M, Spring 2017 Name: ________________________________________________________ 8. Cyclotron (10 points). A cyclotron is a type of particle accelerator. These highenergy particles are also used in nuclear research and in hospitals to obtain radioactive preparations for medical and diagnostic purposes. The cyclotron is made up of two semi-cylinder’s conductors that are opened on the straight side. These D-shaped conductors are made of a non-ferromagnetic material (e.g. , copper) and are called “dees”. The “dees” are placed in a homogenous magnetic field with a magnetic field that is perpendicular to the conductors, as shown in the figure. If an alternating voltage is applied to the dee conductors, and a charged particle enters near the center of the cyclotron (i.e., at point Z in ithe figure), it is accelerated and gains a velocity that runs perpendicular to the magnetic field inside of one of the dees. As we have seen, a magnetic field forces the particles to travel in a circular path. However, the voltage between the “dees” linearly accelerates the charged particles as they cross the gap, increasing their kinetic energy. This effect increases the radius of the circle and so the path is spiral. The high-speed particles exit the cyclotron in the moment when the radius of the path reaches the radius of the cyclotron. . Consider when deuterons, the nucleii of heavy hydrogen, are accelerated in a cyclotron. You can assume that they start with essentially no kinetic energy and that everything stays non-relativistic. (a) Determine the frequency of the voltage source if the value of magnetic field strength in the cyclotron is 1.5 T and the mass of a deuteron is 3.3 x 10-27 kg. (b) Determine the necessary cyclotron radius if we want the deuterons to exit the cyclotron with a kinetic energy of 16 MeV (1 eV = 1.602 x 10-19 Joules). What speed in meters/second does that kinetic energy represent? (c) How many times does the deuteron cross between the “dees”, if the voltage between them is 50 kV? Physics 200, E&M, Spring 2017 
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