A physicist takes some ordinary steel paperclips (which initially do not stick to each other) and observes that he can suspend them in a chain from the bottom of a bar- shaped permanent magnet, as shown below left. After some time has passed, he also observes that he can take away the permanent magnet, and the chain continues to stick together, as shown below right; the steel paperclips have become magnetised. a. Explain both of these observations. (Hint: Steel consists of approximately 98% iron.) b. Each magnetised paperclip weighs 1.1g. If 1.5% of the iron atoms remain aligned after the magnetisation process, calculate the net magnetic moment of one paperclip. The molar mass of iron is mpe = 55.8 g mol- and the magnetic moment of one iron atom is pr. = 2.1 x 10-25 JT- c. The physicist makes an improvised compass by carefully floating the magnetised paperclip on the surface of a cup of water. The strength of the Earth's magnetic field at this location is 40 T and the moment of inertia of the paperclip about its centre is 8.25 x 10 kg m². If the physicist initially places the paperclip onto the water pointing East-North-East (i.e. midway between the directions of East and North-East), calculate the magnitude of the initial angular acceleration of the paperclip in degrees/s? d. If resistive and dissipative forces from the water can be neglected, qualitatively describe the subsequent motion of the paperclip. (Assume the paperclip remains floating on the surface of the water.)
a. Steel consists of approximately 98% iron, which is a ferromagnetic substance. In a non-uniform external magnetic field, a ferromagnetic material is attracted towards the region of greater magnetic field. This explains the first observation. The magnetic dipole moments in a ferromagnetic material can become aligned by an external magnetic field, and then after the external magnetic field is removed, remain partially aligned in regions known as domains. This explains the second observation. b. Assuming the paperclip to be made entirely of iron, the number of atoms in a paperclip can be calculated as (1.1 g)(6.022 x 1024 mol!) = 1.187 X 1022 N = (55.8 g/mol) If 1.5% of the iron atoms remain aligned after the magnetisation process, the net magnetic moment of one paperclip can be calculated as 1.5 100 (1.187 X 10% (2.1 x 10-23) J/T → H = 3.739 x 10-3 J/ T3.7 x 10-*J/T... Assuming the Earth's magnetic field to point towards North, the initial angle between the dipole moment of the paperclip and the direction of the local magnetic field can be calculated as 45 8 = 90° = 67.5 The strength of the Earth's magnetic field at the location of the paperclip is B = 40 x 10-6 T. The torque initially acting on the paperclip can be calculated as T = uB sin 6 = 1.382 10-N m. The moment of inertia of the paperclip about its center is I = 8.25 x 10-kg m? The magnitude of the initial angular acceleration of the paperclip can be calculated as 1.675 rad/s? → a= 95.97 degrees/s2 - 96 degrees/s.d. If resistive and dissipative forces from water can be neglected, the subsequent motion of the paperclip will be oscillatory about the direction pointing towards North. H = (1.5%)Nur →=
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