Problems 𝜇 𝑟 =∞ Problem 1 (See quadrupole figure): A: Why is the pole-tip of an ideal quadrupole an equipotential surface? (10 points) 𝜇 𝑟 =∞ 𝐺= 𝜇 0 𝑁𝐼 𝑎 2 B: Show that the field gradient in the quadrupole is (10 points): Problem 2: Consider the window-frame dipole magnet shown in the figure, with gap width w = 12 cm, gap height g = 2.5 cm and length l = 0.6 m. A: Estimate, how does the magnetic field in the gap change if magnetic permeability of iron changes from 10000 to 1? (30 points) B: Show that the inductance of the magnet is (Hint: use the equation for stored energy in magnetic field and in a coil)(20 points) C: How many ampere-turns (NI) are necessary to achieve a field of 0.5 T in the gap, if the relative permeability of the iron is μr = 4000. What, approximately, is the maximum field in the iron? Where? (20 points) 𝐿= 𝜇 0 𝑁 2 𝑙𝑤 𝑔 D: Air-cooled copper coils carry a maximum current density of about 1.5 A/mm2, while water-cooled copper coils can carry almost 10 times as much. However, water-cooling adds the potential for water leaks and is more expensive, so it is avoided if possible. Would you recommend water-cooled or air-cooled coils for this magnet? How much horizontal space is available between the coils for a beampipe? (20 points) Alexander Plastun

Problems E: If the magnet power supply has a maximum current of 1000 A, how many turns should be used in the coil? (10 points) Problem 3: The permanent magnet is designed to achieve a constant magnetic flux density Bg = 1.0 T in the air gap. The field is to be created by a samarium-cobalt magnet. For the air-gap dimensions of figure find the magnet length lm and the magnet area Am that will achieve the desired air-gap flux density. For the samarium-cobalt assume Bm = 0.45 T, Hm = -375 kA/m. (20 points) Problem 4 (SC magnet): Show that the overlapping current cylinders generate a perfect dipole field in the overlap region, for any value of displacement that is less that 2a0, the diameter of each cylinder. (20 points) Alexander Plastun