Forces and Torques on Current Loops in Magnetic Fields

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. 05

3. Force on each charge Total force on wire Use Force on a wire carrying current I Cross sectional area A, Length L Number of charges/m3 = n Each charge moving at Vavg 06

4. Current- carrying wire in uniform B field Out of page

5. ACT A B C 08

6. ACT A B C 10

7. ACT • What is the force on section d-a of the loop? • Zero • Out of the page • Into the page 12

8. Checkpoint 1a A B C “The net force on any closed loop is zero.” 13

9. Checkpoint 1b y x • In which direction will the loop rotate? • (assume the z axis is out of the page) • Around the x axis • Around the y axis • Around the z axis • It will not rotate 15

10. Checkpoint 1c F R y A B C D E 17

11. Magnetic Dipole Moment Area vector Magnitude = Area Direction uses right hand rule Magnetic Dipole moment 19

12. z turnsmtoward B y m B z x m y B x turnsmtoward B The torque always wants to linemup with B ! 21

13. 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

14. 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

15. Potential energy of a magnetic dipole moment Rotate  from 1 to 2 27

16. U = 0 Lowest U  parallel to B Highest U  antiparallel to B U m B

17. 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

18. 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

19. ACT Consider rotating the dipole to each of the three final orientations shown. qa fa qc B • Does the sign of the work done by you 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

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 the 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