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Sources of the Magnetic Field

Sources of the Magnetic Field. March 22, 2006. Schedule. Watsgoinondisweek?. Today we start looking at how currents create magnetic fields. Quiz on Friday Magnetism Chapter NOT today’s material Next Exam is on April 7 th . New WebAssign on Sources will be posted shortly.

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Sources of the Magnetic Field

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  1. Sources of the Magnetic Field March 22, 2006

  2. Schedule

  3. Watsgoinondisweek? • Today we start looking at how currents create magnetic fields. • Quiz on Friday • Magnetism Chapter • NOT today’s material • Next Exam is on April 7th. • New WebAssign on Sources will be posted shortly.

  4. Last time - Current Loop

  5. Circumference = L/N A length L of wire carries a current i. Show that if the wire is formed into a circular coil, then the maximum torque in a given magnetic field is developed when the coil has one turn only, and that maximum torque has the magnitude … well, let’s see.

  6. Problem continued…

  7. NEW TOPIC: SOURCES OF THE MAGNETIC FIELD Chapter 30

  8. Remember the wire?

  9. Try to remember… Coulomb

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

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

  12. 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!

  13. Using Magnets From the Past

  14. 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 !

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

  16. The Details

  17. Moving right along 1/d Verify this.

  18. A bit more complicatedA finite wire

  19. r q p-q ds P1

  20. More P1

  21. P2

  22. APPLICATION: Find the magnetic field B at point P A Combination of P2 geometries.

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

  24. ds More arc…

  25. The overall field from a circular current loop Sorta looks like a magnet!

  26. Iron

  27. Howya Do Dat?? No Field at C

  28. 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”.

  29. The Calculation

  30. Definition of the Ampere The force acting between currents in parallel wires is the basis for the definition of the ampere, which is one of the seven SI base units. The definition, adopted in 1946, is this: The ampere is that constant current which, if maintained in two straight, parallel conductors of infinite length, of negligible circular cross section, and placed 1 m apart in vacuum, would produce on each of these conductors a force of magnitude 2 x 10-7 newton per meter of length. Huh?

  31. Ampere’s Law The return of Gauss

  32. Remember GAUSS’S LAW?? Surface Integral

  33. 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 the Gauss’ Law of Magnetism! (Sorry)

  34. The next few slides have been lifted from Seb Oliveron the internet Whoever he is!

  35. Biot-Savart • The “Coulombs Law of Magnetism”

  36. 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 • Definition of Ampere

  37. Magnetic Field from a long wire Using Biot-Savart Law r I Take a short vector on a circle, ds B ds Thus the dot product of B & the short vector ds is:

  38. Sum B.ds around a circular path r I B Sum this around the whole ring ds Circumference of circle

  39. Consider a different path • Field goes as 1/r • Path goes as r. • Integral independent of r i

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

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

  42. The Right Hand Rule .. AGAIN

  43. Another Right Hand Rule

  44. COMPARE Line Integral Surface Integral

  45. Simple Example

  46. Field Around a Long Straight Wire

  47. Field INSIDE a WireCarrying UNIFORM Current

  48. The Calculation Graph ????

  49. B R r

  50. 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 WITH AN ATTITUDE!

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