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Start EM Ch.5: Magnetostatics
finish Modern Physics Ch.7: J=L+S
Methods of Math. Physics, Thus. 24 Feb. 2011, E.J. Zita
Magnetostatics:
• Lorentz Force and Biot-Savart Law
• Divergence and Curl of B; Ampere’s Law
Modern Physics
• Spin, Energy levels, Zeeman effect
• Fine structure, Bohr magneton, J=L+S
Ex.5.2: Cycloid motion: B=Bx, E=Ez, find particle motion if it
starts from rest at the origin.
1. Draw. 2. Qualitative analysis. 3. Quantitative analysis.
xˆ
yˆ
v  B  0 dy
dt
B
0
zˆ
dz
dt
0

Ex.5.4: (a) A current I is uniformly distributed over a
wire of circular cross section with radius a. Find J.
Ex.5 (b) Suppose the current density in the wire is
proportional to the distance from the axis: J=ks (for
some constant k). Find the total current in the wire.
Finding field B from current I
• Biot Savart law in general (5.32, p.215)
• Ampere’s law, when symmetry permits (p.221)
Draw Ampere’s law:
Apply Stokes’ Thm. to Magnetostatics


B
d
a




surface

B  dl   0 I
boundary
dI
current density  J 
, so I   J da
da
surface


B
d
a




0 

surface
surface
  B  0 J
J da
Using Ampere’s law
1. Draw, 2. Qualitative analysis, 3. Quantitative
Find B for an infinite uniform surface current K=Kx over the
xy plane. (I=dK/dlength)
Using Ampere’s law
1.Draw, 2. Qualitative analysis, 3. Quantitative
Find B for a solenoid with n closely wound turns per unit length on
a cylinder of radius R and carrying a steady current I.
We’ll finish Ch.5 next week. Choose some HW problems…
Continuing Modern Physics Ch.7:
H atom in Wave mechanics
• Spin, Energy levels
• Zeeman effect
• Fine structure
• Bohr magneton
• J=L+S
H-atom wavefunctions ↔
electron probability distributions:
l = angular momentum wavenumber
Discussion: compare Bohr model to Schrödinger model for H atom.
ml denotes possible orientations of L and Lz (l=2)
Wave-mechanics L ≠ Bohr’s n
Stern-Gerlach showed line splitting, even when l=0.
l = 1, m=0,±1 ✓
l = 0, m=0, s= ±1/2
Normal Zeeman effect
Anomalous Zeeman effect
Magnetic moments
shift energies in B fields
Spin S and orbit L couple to total angular momentum
J = L+ S
Spin-orbit coupling: spin of e- in orbital magnetic field of p
Fine-structure splitting (e.g. 21-cm line)
(Interaction of nuclear spin with electron spin (in an atom) →
Hyper-fine splitting)
Total J + external magnetic field → Zeeman effect
Total J + external magnetic field → Zeeman effect
Total J + external magnetic field → Zeeman effect
History of atomic models:
• Thomson discovered electron, invented plum-pudding model
• Rutherford observed nuclear scattering, invented orbital atom
• Bohr quantized angular momentum, improved H atom model.
• Bohr model explained observed H spectra, derived En = E/n2
and phenomenological Rydberg constant
• Quantum numbers n, l, ml (Zeeman effect)
• Solution to Schrödinger equation shows that En = E/l(l+1)
• Pauli proposed spin (ms= ±1/2), and Dirac derived it
• Fine-structure splitting reveals spin quantum number