Topic 21: Torque and Magnetic Dipoles

Problem 1. A very long length of wire is wrapped into a 100 winding square coil of edge length 0.24 m. The current through the wire is 3.5 A. The coil is oriented in a magnetic field of magnitude 6.8E-3 T as shown above with θ = 33◦. −→ −→ a Verify using the fingers palm thumb RHR, the directions of the depicted forces F1 through F4. Note that Figures (a) and (b) are two views of the same system. b Verify using the curl your fingers around with i RHR, the direction of the magnetic dipole moment of this current loop is the same direction as the normal vector −→n in diagram (b) c Verify the torque τ on the dipole is as depicted by working it out in Figure (b) and then rotating in your mind −→ to Figure (a). Verify that this loop wants to turn to align with B . d Calculate the magnitude of the magnetic dipole moment of the coil (units: A·m2 e Calculate the magnitude of the torque (τ) acting on this coil? f By how much work would we need to do to twist this coil into its maximum potential energy orientation.

Problem 2. The ﬁgure shows a circular current loop in the yz plane. The current ﬂows out of the page (+z) at the top and into the page at the bottom. What isn ˆ? x z i

Problem 3. A 100-turn rectangular coil of side lengths 10 cm and 20 cm lies in the xy plane and carries a counterclockwise current of 5 A. There is a uniform magnetic ﬁeld B~ = (+2ˆı − 3kˆ) T. (a) What is the (vector) torque on the coil? (b) What axis does that torque tend to rotate the coil around? Problem 4. A current-carrying coil with magnetic dipole moment 6 J/T makes an angle of 120 degrees with a 2 T magnetic ﬁeld. By allowing the coil to rotate to its minimum energy orientation, you use the coil’s energy to do work. Can the coil lift a 2 kg weight one meter before using up all its potential energy?

1 Problem 5. The ﬁgure shows a rectangular, 26-turn coil of wire, of dimensions 14 cm by 5.3 cm. It carries a current of 0.10 A and is hinged along one long side. It is mounted in the xy plane, at an angle of 30 degrees to the direction of a uniform magnetic ﬁeld of magnitude 0.44 T. Find the (a) x, (b) y, and (c) z components of the torque acting on the coil about the hinge line.

Problem 6. The ﬁgure shows a wood cylinder of mass m and length L, with N turns of wire wrapped around it longitudinally, so that the plane of the wire coil contains the long central axis of the cylinder. The cylinder is released on a plane inclined at an angle to the horizontal, with the plane of the coil parallel to the incline plane. If there is a vertical uniform magnetic ﬁeld of magnitude B, what is the least current i through the coil that keeps the cylinder from rolling down the plane? Express your answer in terms of the variables given and the gravitational acceleration g.

Problem 7. The ﬁgure gives the potential energy U of a magnetic dipole in an −→ −→ external magnetic ﬁeld B , as a function of angle φ between the directions of B and the dipole moment. The vertical axis scale is set by Us = 1.9E-4 J. The dipole can be rotated about an axle with negligible friction so as to change φ. Counterclockwise rotation fromφ = 0 yields positive values of φ, and clockwise rotations yield negative values. The dipole is to be released at angle φ = 0 with a rotational kinetic energy of 6.7E-4 J, so that it rotates counterclockwise. To what maximum value of φ will it rotate?

ANSWERS

1. (a-c) just verify ; (d) 20.16 Am2 (e) 0.0747 Nm (f) 0.252 J 2. Right 3. (a) 10kˆ (b) Towards equilibrium - towards parallel to B 4. No it cannot lift it. 5. (a) 0 (b) -0.0074 Nm (c) 0 Nm 6. i = mg/(2NLB) 7. 114.24 degrees

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