Question:

Consider the loop of wire in Figure 28.25a. Imagine it is pivoted along side 4, which is parallel to the z axis and fastened so that side 4 remains fixed and the rest of the loop hangs vertically in the gravitational field of the Earth but can rotate around side 4 (Fig. 28.25b). The mass of the loop is 50.0 g, and the sides are of lengths a = 0.200 m and b = 0.100 m. The loop carries a current of 3.50 A and is immersed in a vertical uniform magnetic field of magnitude 0.010 0 T in the positive y direction (Fig. 28.25c). What angle does the plane of the loop make with the vertical? Figure 28.25 (a) The dimensions of a rectangular current loop. (b) Edge view of the loop sighting down sides (2) and (4). (c) An edge view of the loop in (b) rotated through an angle with respect to the horizontal when it is placed in a magnetic field. The magnetic torque causes the loop to rotate in a clockwise The loop hangs vertically direction around side 4, whereas and is pivoted so that it the gravitational torque is in the can rotate around side 4. opposite direction. cos 8 2 4 sin e 3 g (2) (2 . b a b CSolution CON ceptualize 1n the edge view of gure 28.25b, notice that the magnetic moment of the loop is to the left. Therefore, when the loop is in the magnetic eld, the magnetic torque on the loop causes it to rotate in a clockwise direction around side @, which we choose as the rotation axis. Imagine the loop making this clockwise rotation so that the plane of the loop is at some angle 6 to the vertical as in gure 28.25c. The gravitational force on the loop exerts a torque that would cause a rotation in the counterclockwise direction if the magnetic eld were turned off. categorize At some angle of the loop, the two torques described in the Conceptualize step are equal in magnitude and the loop is at rest. We therefore model the loop as a rigid object in equilibrium. Analyze Evaluate the magnetic torque on the loop ?B : j x 3 : ,u.Bsin (900 _ (9)1\": : IABcos 9 E : IabB cos 9 1': about side Gt) from muation 28.18: Evaluate the gravitational torque on the loop about ;> : > > b . 1' X mg :mgsmdk side Gt), noting that the gravitational force can be g 2 modeled to act at the center of the loop: From the rigid body in equilibrium model, add the Z ;> : IribBcos 9 I} + my gain 6 T: : 0 torques and set the net torque equal to zero: 2 Solve for 9: 1,3530% 9 _ mg b sin 9 i tan 6 2MB 2 my 9 = tan1 2MB mg H 1 W _ an i (0.0500kg){9.80m/s2} Substitute numerical values: _ 1.64\" Finalize The angle is relatively small, so the loop still hangs almost vertically. If the current I or the magnetic eld B is increased, however, the angle increases as the magnetic torque becomes stronger. \fTorque " ,: Consider the loop of wire in Figure a. Imagine it is pivoted alongside v, which is parallel to the z axis and fastened so that side V remains fixed and the rest of the loop hangs vertically in the gravitational field of the Earth but can rotate around side v (Fig. b). The mass of the loop is 62.0 g, and the sides are of lengths a = 0.200 rm and b = 0.100 m. The loop carries a (4;) G) '1 1' _) 5;) current of 3.30 A and is immersed in a vertical I a J: 41\" . g cm 9 y a 11 '2 ' uniform magnetic field of magnitude 0.010 T in f? @ E ' |_X 351\" 9 |_x the positive y direction (Fig. c). What angle does @1. ii @ 175' 5339 E @I the plane of the loop make with the vertical? l) Warning* make sure to go through all steps as shown adding justification for the step in a single sentence