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Biot-Savart Law

Biot-Savart Law. Single Charge:. The Biot-Savart law for a short length of thin wire. Current:. Moving charge produces a curly magnetic field. B units: T (Tesla) = kg s -2 A -1. Magnetic Field of Current Distributions. Four-step approach:.

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Biot-Savart Law

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  1. Biot-Savart Law Single Charge: The Biot-Savart law for a short length of thin wire Current: Moving charge produces a curly magnetic field B units: T (Tesla) = kg s-2A-1

  2. Magnetic Field of Current Distributions Four-step approach: • Cut up the current distribution into pieces and draw B • Write an expression for B due to one piece • Add up the contributions of all the pieces • Check the result

  3. A Long Straight Wire Step 1: Cut up the current distribution into pieces and draw B. Origin: center of wire Vector r: Magnitude of r: Unit vector:

  4. A Long Straight Wire : Step 2: Write an expression for B due to one piece. B field due to one piece:

  5. A Long Straight Wire need to calculate only z component

  6. A Long Straight Wire Step 3: Add up the contribution of all the pieces.

  7. A Long Straight Wire Special case:x<<L What is the meaning of “x”?

  8. A Long Straight Wire Step 4: Check results  direction far away: r>>L   units:

  9. Semi-infinite Straight Wire For Infinite Wire For Semi-Infinite Wire 0 Even Function: Half the integral …

  10. Off-axis for Long Straight Wire y a Angle between Rewrite in terms of x See Quest Course Resources for details (offaxisline.pdf)

  11. Right-hand Rule for Wire Conventional Current Direction

  12. Question Current carrying wires below lie in X-Y plane.

  13. Question T

  14. Magnetic Field of a Wire Loop Step 1: Cut up the distribution into pieces Make use of symmetry! Need to consider only Bz due to one dl

  15. Magnetic Field of a Wire Loop Step 2: B due to one piece Origin: center of loop Vector r: Magnitude of r: Magnetic field due to one piece: Unit vector: l:

  16. Magnetic Field of a Wire Loop Step 2: B due to one piece need only z component:

  17. Magnetic Field of a Wire Loop Step 3: Sum the contributions of all pieces Magnetic field of a loop along its axis:

  18. Magnetic Field of a Wire Loop Step 4: Check the results units:  direction:  Check several pieces with the right hand rule Note: We’ve not calculated or shown the “rest” of the magnetic field

  19. Magnetic Field of a Wire Loop Special case: center of the loop Using general form (z=0) :

  20. Magnetic Field of a Wire Loop Special case: far from the loop for z>>R: The magnetic field of a circular loop falls off like 1/z3

  21. Magnetic Field of a Semicircle Special case: at center of the semicircle For whole loop What is for 1.5 loops?

  22. A Coil of Wire single loop: What if we had a coil of wire? For N turns:

  23. Magnetic Dipole Moment  - vector in the direction of B far from coil: far from dipole: magnetic dipole moment:

  24. Twisting of a Magnetic Dipole In the presence of external magnetic field a current-carrying loop rotates to align the magnetic dipole moment  along the field B. The magnetic dipole moment  acts like a compass needle!

  25. Exercise: a loop of radius R and a long straight wire. The center of the loop is 2R from the wire. I X I What are the directions of the magnetic fields at the center of the loop? What is the net magnetic field at the center of the loop?

  26. The Magnetic Field of a Bar Magnet N S How does the magnetic field around a bar magnet look like?

  27. Magnets and Matter How do magnets interact with each other? Magnets interact with iron or steel, nickel, cobalt. Does it interact with charged tape? Does it work through matter? Does superposition principle hold? • Similarities with E-field: • can repel or attract • superposition • works through matter • Differences with E-field: • B-field only interacts with some objects • curly pattern • only closed field lines

  28. Magnetic Field of Earth The magnetic field of the earth has a pattern that looks like that of a bar magnet Horizontal component of magnetic field depends on latitude Maine: ~1.5.10-5 T Texas: ~2.5x10-5T Can use magnetic field of Earth as a reference to determine unknown field.

  29. Magnetic Monopoles N N S S An electric dipole consists of two opposite charges – monopoles Break magnet: There are no magnetic monopoles!

  30. The Atomic Structure of Magnets S N The magnetic field of a current loop and the magnetic field of a bar magnet look the same. Electrons What is the direction? One loop: What is the average current I? current=charge/second:

  31. Magnetic Dipole Moment Magnetic dipole moment of 1 atom: Method 1: use quantized angular momentum Orbital angular momentum: Quantum mechanics:L is quantized: If n=1:

  32. Magnetic Dipole Moment Magnetic dipole moment of 1 atom: Method 2: estimate speed of electron Momentum principle: Circular motion: • – angular speed

  33. Magnetic Dipole Moment Magnetic dipole moment of 1 atom: 6.1023 atoms Mass of a magnet:m~5g Assume magnet is made of iron:1 mole – 56 g number of atoms = 5g/56g . 6.1023 ~ 6.1022

  34. Modern Theory of Magnets 1. Orbital motion There is no ‘motion’, but a distribution Spherically symmetric cloud (s-orbital) has no  Only non spherically symmetric orbitals (p, d, f) contribute to  There is more than 1 electron in an atom

  35. Modern Theory of Magnets 2. Spin Electron acts like spinning charge - contributes to  Electron spin contribution to  is of the same order as one due to orbital momentum Neutrons and proton in nucleus also have spin but their ‘s are much smaller than for electron same angular momentum: NMR, MRI – use nuclear 

  36. Modern Theory of Magnets Why are only some materials magnetics? Alignment of atomic magnetic dipole moments: ferromagnetic materials: iron, cobalt, nickel most materials

  37. Modern Theory of Magnets Magnetic domains Hitting or heating while in a magnetic field can magnetize the iron Hitting or heating can also demagnetize

  38. Why are there Multiple Domains? Magnetic domains

  39. Iron Inside a Coil Multiplier effect: Electromagnet:

  40. Magnetic Field of a Solenoid B Step 1:Cut up the distribution into pieces Step 2:Contribution of one piece origin:center of the solenoid one loop: Number of loops per meter:N/L Number of loops in z: (N/L) z Field due to z:

  41. Magnetic Field of a Solenoid B Step 3:Add up the contribution of all the pieces Magnetic field of a solenoid:

  42. Magnetic Field of a Solenoid Special case: R<<L, center of the solenoid: in the middle of a long solenoid

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