1. Magnetic Fields Due to Currents
Chapter 29

2. Remember the wire?

3. Try to remember…
Coulomb

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

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

6. 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!

7. From the Past
Using Magnets

8. 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 !

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

10. The Details

11. Moving right along
1/d
Verify this.

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

13. More arc… ds

14. The overall field from a circular current loop

15. Iron

16. Howya Do Dat??
No Field at C

17. 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”.

18. The Calculation

19. Invisible Summary Biot-Savart Law Force between two wires
(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

20. Ampere’s Law
The return of Gauss

21. Remember GAUSS’S LAW??
Surface
Integral

22. 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!

23. AMPERE’S LAW by SUPERPOSITION:
We will do a LINE INTEGRATION
Around a closed path or LOOP.

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

25. The Right Hand Rule .. AGAIN

26. Another Right Hand Rule

27. COMPARE
Line Integral
Surface Integral

28. Simple Example

29. Field Around a Long Straight Wire

30. Field INSIDE a Wire Carrying UNIFORM Current

31. The Calculation

32. R r B

33. 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

34. #79 The figure below shows a cross section of an infinite conducting sheet carrying a current per unit x-length of l; the current emerges perpendicularly out of the page. (a) Use the Biot–Savart law and symmetry to show that for all points P above the sheet, and all points P´ below it, the magnetic field B is parallel to the sheet and directed as shown. (b) Use Ampere's law to find B at all points P and P´.

35. FIRST PART
Vertical Components
Cancel

36. Apply Ampere to Circuit
Infinite Extent B

37. The “Math”
Bds=0
Infinite Extent B
Distance not a factor!

38. A Physical Solenoid

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

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

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

42. Another Way

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

44. Two Coils

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

46. The Toroid
Slightly less
dense than
inner portion

47. The Toroid

48. 15. A wire with current i=3. 00 A is shown in Fig. 29-46
15.  A wire with current i=3.00 A is shown in Fig 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?

49. 38.  In Fig , 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?