1. Prof. David R. Jackson ECE Dept. Spring 2016 Notes 33 ECE 3318 Applied Electricity and Magnetism 1

2. Force on Wire Single charge: Wire: q B v F B I F 2 (derivation omitted)

3. Force on Wire (cont.) Force on wire: The contour C is in the direction of the reference direction for the current. B I F C 3 Note: There is no total force on a wire due to the magnetic field produced by itself.

4. Example Find the force F 2 on wire 2 4 I2I2 I1I1 1 2 z h z = 0 z = L x y Assume that the wires are long compared to the separation. Note: The force on wire 2 comes from the magnetic field due to wire 1.

5. 5 The contour C 2 runs from point A to point B (left to right), defined by the current reference direction on wire 2. I2I2 I1I1 1 2 z h z = 0 z = L x C2C2 y A B Example (cont.)

6. 6 F 1 = - F 2 F2F2 A B I1I1 I2I2 z x z = L Note: The contour C 2 runs from z = 0 to z = L (since I 2 runs in this direction).

7. Definition of Amp Assume F1F1 F2F2 A B I I z x 2 1 7 Definition of Amp: Note:

8. Definition of Amp (cont.) (an exact constant) 8 Force per unit length: so

9. Example 9 F1F1 F2F2 A B I I z x 2 1 (attractive force) In a power substation, the current in two parallel buses is 1 [ kA ]. The buses are 1 [ m ] apart. What is the force per unit length between the two buses?

10. Example (cont.) 10 Note: 1 [kG]  9.8 [N] = 2.2 [lbs] or

11. Example 11 During a lighting strike, the current in two parallel wires inside the wall of a house reaches 10 [ kA ]. The wires are 1 [ cm ] apart. What is the force per unit length between the two wires? (attractive force) F1F1 F2F2 A B I I z x 2 1

12. Example (cont.) 12 or Note: 1 [kG]  9.8 [N] = 2.2 [lbs]

13. Example 13 In a DC motor, the armature consists of N = 10,000 turns (loops) of wire, with each turn being L = 0.1 [ m ] in length (in the direction parallel to the axis). The magnetic flux density produced by the stator is B = 0.5 [ T ]. The radius of the armature is R = 0.05 [ m ]. Find the torque on the armature. The current through the motor is 3 [ A ]. Assume that the magnetic field is constant and perpendicular to the wire current. Note: There is no net force on the entire loop, since the force from opposite sides cancels. Both sides (left ad right) of the loop contribute equally to the torque. Axis of rotation + on right edge, - on left edge For single turn:

14. Example (cont.) 14 Therefore, we have or Axis of rotation

15. Example (cont.) 15 We could also write the answer for the torque as follows: Axis of rotation or

16. Torque on Current Loop 16 A planar current loop of wire with an arbitrary shape carries a current I. B = magnetic flux density vector (assumed constant here). Define magnetic dipole moment of loop: Torque vector on loop: (arbitrary direction)

17. DC Motor 17 A loop rotating in a DC magnetic field is shown below. rr N N S N N S A commutator is needed to reverse the current every 180 o and make the torque in same direction. http://en.wikipedia.org/wiki/DC_motor

18. DC Motor (cont.) 18 http://en.wikipedia.org/wiki/Commutator_(electric) The commutator reverses the loop current every 180 o. Note: The rings are positioned so that the torque changes sign (current reverses) at the point where the torque is zero. Commutator (rings) Brushes

19. DC Motor (cont.) 19 In practice, there are multiple loops and commutator segments. The torque is thus more constant as the armature turns. http://www.mmsonline.com/columns/gaging-commutators

20. Force from a Magnet 20 rr N S N N Large block of iron (base area) The magnetic field is almost constant inside the air gap region, since the iron has a high permeability. Assume most of the stored energy is inside the gap region.

21. Force from a Magnet (cont.) 21 rr N S N N Large block of iron (base area) Principle of “virtual work”: We then have: