1. Biot-Savart Law Moving charge produces a curly magnetic field
Single Charge:
The Biot-Savart law for a short length of thin wire
Current:
B units: T (Tesla) = kg s-2A-1
𝜇 0 4𝜋 = 10 −7 T∙m A

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
Has same distance dependence as E-field from long uniformly charged wire.
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 −∞ +∞ ∞
𝐵 ∞ = 𝜇 0 4𝜋 2𝐼 𝑥
For Semi-Infinite Wire +∞
Half the integral for the infinite wire … so half the result.
Even Function: Half the integral …
𝐵 𝑠𝑒𝑚𝑖 = 𝜇 0 4𝜋 𝐼 𝑥 −∞

10. Off-axis for Long Straight Wirey a
Angle between
∆ 𝑙 𝑟
∆ 𝐵 = 𝜇 0 4𝜋 1 𝑟 2 ∆𝑦 sin 𝛼 − 𝑧
Rewrite ∆𝑦 in terms of ∆𝛼 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. +z

13. Question 𝐵 𝑤𝑖𝑟𝑒 = 𝐵 𝑒𝑎𝑟𝑡ℎ tan⁡(𝜃) =(2×1 0 −5 T) tan⁡(12°)B
𝐵 𝑤𝑖𝑟𝑒 = 𝐵 𝑒𝑎𝑟𝑡ℎ tan⁡(𝜃)
=(2×1 0 −5 T) tan⁡(12°)
=4.3×1 0 −6 T

14. Magnetic Field of a Wire Loop
Step 1:
Cut up the distribution into pieces
Note sense of dl-vector.
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:
Expression for delta-l is from Taylor expansion.
Note that angle between dl and r is a right angle. Can prove by showing cross product is dl * r * sin(theta) = dl * r, so theta = 90.
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
Magnetic field of a loop:

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

21. 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

22. Magnetic Field of a Semicircle
Special case: only 𝐵 𝑧 at center of the semicircle
For whole loop
0 𝜋 = 𝜋
𝐵 𝑧,𝑠𝑒𝑚𝑖 = 𝜇 0 4𝜋 𝜋𝐼 𝑅
𝐵 𝑧,∆𝜃 = 𝜇 0 4𝜋 2𝜋𝐼 𝑅 ∆𝜃 2𝜋
What is ∆𝜃for 1.5 loops?

23. A Coil of Wire single loop: What if we had a coil of wire?
For N turns:
Make a usual coil – and then with loops running in opposite directions.

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

25. Exercise
What is the magnetic dipole moment  of a 3000-turn 35 cm rectangular coil that carries a current of 2 A?
for one turn
for N turns
Equation is ~valid even if loop is not exactly circular

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

27. Exercise: a loop of radius R and a long straight wire
Exercise: a loop of radius R and a long straight wire. The center of the loop is 2R from the wire. I
𝐵 𝑤𝑖𝑟𝑒 = 𝜇 0 4𝜋 2𝐼 𝑟
𝐵 𝑙𝑜𝑜𝑝 = 𝜇 0 4𝜋 2𝜋𝐼 𝑅 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?
𝐵 𝑙𝑜𝑜𝑝 − 𝐵 𝑤𝑖𝑟𝑒 = 𝜇 0 4𝜋 2𝜋𝐼 𝑅 − 𝜇 0 4𝜋 2𝐼 2𝑅
= 𝜇 0 𝐼 4𝜋𝑅 2𝜋−1

28. The Magnetic Field of a Bar Magnet
How does the magnetic field around a bar magnet look like? N S
Do experiment with compass and bar magnet to figure out.

29. 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
Superposition – take 2 magnets and try to compensate the field detected by compass
Differences with E-field:
B-field only interacts with some objects
curly pattern
only closed field lines

30. 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: ~ T
Texas: ~2.5x10-5 T
S-N are mixed up because of compass!
Magnetic poles are 1300 km away from actual poles
Can use magnetic field of Earth as a reference to determine unknown field.

31. Current is flowing to the right in a wire
Current is flowing to the right in a wire. The magnetic field at the position P points
Current is flowing to the right in a wire. The magnetic field at the position P points A. B. C. D.
Choice Five
Choice Six
Choice Four
Choice Three
Choice Two
Choice One

32. What is the direction of the magnetic field inside the solenoid?
Current upward on side nearest you A. B. D. C.
Choice Five
Choice Six
Choice Four
Choice Three
Choice Two
Choice One

33. Current clockwise; north pole on top
A current in the loop has created the magnetic field, B, shown below. What is the current direction in this loop if you look from the top? And which side of the loop is the north pole? (To get the pole, you need to replace the loop with a bar magnet that has the same field direction)
Current clockwise; north pole on top
Current counterclockwise, north pole on top
Current clockwise; north pole on bottom
Current counterclockwise, north pole on bottom B C

34. Magnetic Monopoles
An electric dipole consists of two opposite charges – monopoles
Break magnet: N S S N
Do experiment: attach two magnets S_N-S_N – act juts like one SN magnet. Break it – got two magnets.
There are no magnetic monopoles!

35. The Atomic Structure of Magnets
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? S N
current=charge/second:
Also – distnce dependence 1/r3.

36. Magnetic Dipole Moment
Magnetic dipole moment of 1 atom:
Method 1: use quantized angular momentum
Orbital angular momentum:
Quantum mechanics: L is quantized:
Also assume only 1 electron in one atom
If n=1:

37. Magnetic Dipole Moment
Magnetic dipole moment of 1 atom:
Method 2: estimate speed of electron
Momentum principle:
Circular motion:
– angular speed
R = 0.5 x 10^-10 meters.

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

39. 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

40. 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 

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

42. Modern Theory of Magnets
Magnetic domains
Alloys – prevent iron from disordering after being magnetized.
Hitting, heating above critical T – destroys order
Hitting or heating while in a magnetic field can magnetize the iron
Hitting or heating can also demagnetize

43. Why are there Multiple Domains?
Magnetic domains

44. Iron Inside a Coil
Multiplier effect:
Electromagnet:

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

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

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