Physics 212 Lecture 13

1. Physics 212 Lecture 13 Forces and Torques on Currents

2. Key Concepts: • Forces & Torques on loops of current due to a magnetic field. • The magnetic dipole moment. Today’s Plan: • Review of cross product • Forces & Torques • Magnetic dipole moment • Example problem 05

3. Last Time: force on charge This Time: force on wire z y F B I x 06

4. ACT A square loop of wire is carrying current in the counterclockwise direction. There is a horizontal uniform magnetic field pointing to the right. What is the force on section a-b of the loop? A. Zero B. Out of the page C. Into the page 08

5. ACT A square loop of wire is carrying current in the counterclockwise direction. There is a horizontal uniform magnetic field pointing to the right. What is the force on section b-c of the loop? A. Zero B. Out of the page C. Into the page 10

6. ACT A square loop of wire is carrying current in the counterclockwise direction. There is a horizontal uniform magnetic field pointing to the right. What is the force on section d-a of the loop? A. Zero B. Out of the page C. Into the page 12

7. Checkpoint 1a “The net force on any closed loop is zero.” Check simulations if in doubt. A square loop of wire is carrying current in the counterclockwise direction. There is a horizontal uniform magnetic field pointing to the right. What is the direction on the net force on the loop? A. Out of the page B. Into of the pageC. The net force on the loop is zero 13

8. Checkpoint 1b A square loop of wire is carrying current in the counterclockwise direction. There is a horizontal uniform magnetic field pointing to the right. y x In which direction will the loop rotate (assume the z axis is out of the page)? A. Around the x axis B. Around the y axisC. Around the z axis D. It will not rotate 15

9. Checkpoint 1c F R A square loop of wire is carrying current in the counterclockwise direction. There is a horizontal uniform magnetic field pointing to the right. y What is the direction of the net torque on the loop? A. Up B. Down C. Out of the page D. Into the pageE. The net torque is zero 17

10. Magnetic Dipole Moment Area vector Magnitude = Area Direction uses R.H.R. Magnetic Dipole moment 19

11. mMakes Torque Easy! z turnsmtoward B y m B z x m y B x turnsmtoward B The torque always wants to linemup with B ! 21

12. Practice with mand t I m z is up (turnsmtoward B) y m B x B In this casemis out of the page (using right hand rule) 22

13. Checkpoint 2a Biggest when Three different orientations of a magnetic dipole moment in a constant magnetic field are shown below. Which orientation results in the largest magnetic torque on the dipole? 24

14. Magnetic Field can do Work on Current Define U = 0 at position of maximum torque m B From Physics 211: From Physics 212: 27

15. Checkpoint 2b q f U = -mBcosq U = +mBcosf Three different orientations of a magnetic dipole moment in a constant magnetic field are shown below. Which orientation has the most potential energy? U = 0 30

16. ACT Three different orientations of a magnetic dipole moment in a constant magnetic field are shown below. We want to rotate the dipole in the CCW direction. qa fa qc B • First, consider rotating to position c. What are the signs of the work done by you and the work done by the field? • Wyou > 0, Wfield > 0 • Wyou > 0, Wfield < 0 • Wyou < 0, Wfield > 0 • Wyou < 0, Wfield < 0 • DU > 0, so Wfield < 0. Wyou must be opposite Wfield • Also, torque and displacement in opposite directions  Wfield < 0 30

17. ACT Three different orientations of a magnetic dipole moment in a constant magnetic field are shown below. Consider rotating the dipole to each of the three final orientations shown. qa fa qc B • Do the signs depend on which position (a, b, or c) the dipole is rotated to? • Yes • No The lowest potential energy state is with dipole parallel to B. The potential energy will be higher at any of a, b, or c. 30

18. Checkpoint 2c Three different orientations of a magnetic dipole moment in a constant magnetic field are shown below. In order to rotate a horizontal magnetic dipole to the three positions shown, which one requires the most work done by the magnetic field? qa fa qc B 30

19. Checkpoint 2c (c): (b): (a): Three different orientations of a magnetic dipole moment in a constant magnetic field are shown below. In order to rotate a horizontal magnetic dipole to the three positions shown, which one requires the most work done by the magnetic field? qa fa qc B 30

20. Calculation z A square loop of side a lies in the x-zplane with current I as shown. The loop can rotate about x axis without friction. A uniform field B points along the +z axis. Assume a, I, and B are known. How much does the potential energy of the system change as the coil moves from its initial position to its final position. z . B B 30˚ y y a I x final initial • Conceptual Analysis • A current loop may experience a torque in a constant magnetic field • t = m X B • We can associate a potential energy with the orientation of loop • U= -m ∙ B • Strategic Analysis • Find m • Calculate the change in potential energy from initial to final 32

21. Calculation • What is the direction of the magnetic moment of this current loop in its initial position? (A) +x(B) -x(C) +y (D) -y . z z X m ● y x y Right Hand Rule z A square loop of side a lies in the x-zplane with current I as shown. The loop can rotate about x axis without friction. A uniform field B points along the +z axis. Assume a, I, and B are known. z . B B 30˚ y y a I x final initial 34

22. Calculation • What is the direction of the torque on this current loop in the initial position? (A) +x(B) -x(C) +y (D) -y . z X B m y z A square loop of side a lies in the x-zplane with current I as shown. The loop can rotate about x axis without friction. A uniform field B points along the +z axis. Assume a, I, and B are known. z . B B 30˚ y y a I x final initial 36

23. Calculation • What is the potential energy of the initial state? (A) Uinitial < 0 (B) Uinitial = 0 (C) Uinitial > 0 z B q m y z A square loop of side a lies in the x-zplane with current I as shown. The loop can rotate about x axis without friction. A uniform field B points along the +z axis. Assume a, I, and B are known. z . B B 30˚ y y a I x final initial 38

24. Calculation • What is the sign of the potential energy in the final state? (A) Ufinal < 0 (B) Ufinal = 0 (C) Ufinal > 0 z z initial final Energy must increase ! B B q = 90o + 30o q = 90o y y m m z A square loop of side a lies in the x-zplane with current I as shown. The loop can rotate about x axis without friction. A uniform field B points along the +z axis. Assume a, I, and B are known. z . B B 30˚ y y a I x final initial Check: m moves away from B 40

25. Calculation z B q = 120o y m z A square loop of side a lies in the x-zplane with current I as shown. The loop can rotate about x axis without friction. A uniform field B points along the +z axis. Assume a, I, and B are known. z . B B 30˚ y a I x final initial • What is the potential energy of the final state? (A) (B) (C) 44