Start EM Ch.5: Magnetostaticsfinish 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.

ˆ ˆ ˆ

0

0 0

dy dzdt dt

B

x y z

v B

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

0

0

0

current density , so

surface boundary

surface

surface surface

B d B d I

dIJ I J d

da

B d J d

B J

a l

a

a a

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 pFine-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