Magnetic Fields Due to Currents

1. Magnetic Fields Due to Currents Chapter 29

2. Remember the wire?

3. The “Coulomb’s Law” of Magnetism The Law of Biot-Savart A Vector Equation

4. For the Magnetic Field,current “elements” create the field. This is the Law of Biot-Savart This is to calculate B!

5. Magnetic Field of a Straight Wire • We intimated via magnets that the Magnetic field associated with a straight wire seemed to vary with 1/d. • We can now PROVE this!

6. Using Magnets From the Past

7. Directions: The Right Hand Rule Right-hand rule: Grasp the element in your right hand with your extended thumb pointing in the direction of the current. Your fingers will then naturally curl around in the direction of the magnetic field lines due to that element. Reminder !

8. Let’s Calculate the FIELD Note: For ALL current elements in the wire: ds X r is into the page

9. The Details

10. Moving right along 1/d Verify this.

11. Center of a Circular Arc of a Wire carrying current

12. ds More arc…

13. The overall field from a circular current loop

14. Iron

15. Howya Do Dat?? No Field at C

16. Force Between Two Current Carrying Straight Parallel Conductors Wire “a” creates a field at wire “b” Current in wire “b” sees a force because it is moving in the magnetic field of “a”.

17. The Calculation

18. Invisible Summary • Biot-Savart Law • (Field produced by wires) • Centre of a wire loop radius R • Centre of a tight Wire Coil with N turns • Distance a from long straight wire • Force between two wires

19. Ampere’s Law • The return of Gauss

20. Remember GAUSS’S LAW?? Surface Integral

21. Gauss’s Law • Made calculations easier than integration over a charge distribution. • Applied to situations of HIGH SYMMETRY. • Gaussian SURFACE had to be defined which was consistent with the geometry. • AMPERE’S Law is compared to Gauss’ Law for Magnetism!

22. AMPERE’S LAWby SUPERPOSITION: We will do a LINE INTEGRATION Around a closed path or LOOP.

23. Ampere’s Law USE THE RIGHT HAND RULE IN THESE CALCULATIONS

24. The Right Hand Rule .. AGAIN

25. Another Right Hand Rule

26. COMPARE Line Integral Surface Integral

27. Simple Example

28. Field Around a Long Straight Wire

29. Field INSIDE a WireCarrying UNIFORM Current

30. The Calculation

31. B R r

32. Procedure • Apply Ampere’s law only to highly symmetrical situations. • Superposition works. • Two wires can be treated separately and the results added (VECTORIALLY!) • The individual parts of the calculation can be handled (usually) without the use of vector calculations because of the symmetry. • THIS IS SORT OF LIKE GAUSS’s LAW

33. A Physical Solenoid

34. Inside the Solenoid For an “INFINITE” (long) solenoid the previous problem and SUPERPOSITION suggests that the field OUTSIDE this solenoid is ZERO!

35. More on Long Solenoid Field is ZERO! Field is ZERO Field looks UNIFORM

36. The real thing….. Finite Length Weak Field Stronger Fairly Uniform field

37. Another Way

38. Application • Creation of Uniform Magnetic Field Region • Minimal field outside • except at the ends!

39. Two Coils

40. “Real” Helmholtz Coils Used for experiments. Can be aligned to cancel out the Earth’s magnetic field for critical measurements.

41. The Toroid Slightly less dense than inner portion

42. The Toroid

43. 15.  A wire with current i=3.00 A is shown in Fig.29-46. Two semi-infinite straight sections, both tangent to the same circle, are connected by a circular arc that has a central angle θ and runs along the circumference of the circle. The arc and the two straight sections all lie in the same plane. If B=0 at the circle's center, what is θ?

44. 38.  In Fig. 29-64, five long parallel wires in an xy plane are separated by distance d=8.00 cm , have lengths of 10.0 m, and carry identical currents of 3.00 A out of the page. Each wire experiences a magnetic force due to the other wires. In unit-vector notation, what is the net magnetic force on (a) wire 1, (b) wire 2, (c) wire 3, (d) wire 4, and (e) wire 5?