Lecture 18

Circular orbit in uniform magnetic field

Assume magnetic field B ^→ uniform.
Electron of charge q = - e launched with v ^→ perpendicular to field lines
⇒ θ = 90^∘
⇒ force on electron: F = |q v B| = e v B
v = speed, B magnitude of magnetic field
Direction of F ^→ given by right-hand rule
Radius r proportional to speed v:
r = ‎m v‏‎q B‏
The larger B, the smaller the radius.

Mass spectrometer

Ion source → velocity selector → magnetic sector → detector

Velocity selector:
1. To get through slits s₁ and s₂, ion (charge q) must fly straight!
2. Straight path through crossed electric ( E ^→) and magnetic ( B ^→) fields?
3. Balance:
  electrostatic ↔ magnetic force
  Note: both forces are vertical!
  
  ⇒ q E = q v B
4. Required speed:
  v = ‎E‏‎B‏

Mass spectrometer

Ion source → velocity selector → magnetic sector → detector

Magnetic sector:
1. No electric field, but uniform magnetic field B ^→'
2. ⇒ Ion describes circular arc in a plane perpendicular to field lines
3. Radius r (derived earlier):
  r = ‎m v‏‎q B'‏
Solve for mass m, substitute v=E/B:
m = ‎q B'‏‎v‏ r = ‎q B B'‏‎E‏ r

Magnetic field of a long straight wire

Example for non-uniform magnetic field:
circular field lines around current-carrying wire.
Direction of the field lines:
use right-hand rule
Experimental result for B:
given I = current through wire,
B = μ₀ ‎I‏‎2 π r‏
at a distance r from the wire.

Proportionality constant:
μ₀ = 4 π × 10^‎−7‏ ‎T m‏‎A‏
μ₀ = "permeability of free space"

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Force between two wires

Wire 1 creates field B:
B = μ₀ ‎I₁‏‎2 π r‏ at a distance r from the wire.
Here, I₁ = current through wire 1.
Wire 2 experiences force F due to wire 1:
F = I₂ B L₂
Here, L₂ = length of wire 2, and I₂ = current through wire 2.
Put the expression for B into the equation for F:
F = μ₀ ‎1‏‎2 π r‏ I₁ I₂ L₂

I₁ and I₂ parallel ⇒ F positive ⇒ wires attract.
I₁ and I₂ anti-parallel ⇒ F positive ⇒ wires repel.

Magnetic fields and superposition principle

Magnetic fields can be generated by (e.g.):
1. "permanent magnets"
2. wires carrying electric current
3. any kind of moving charges (such as an electron beam)
What is the magnetic field when more than one source is present, e.g., two bar magnets?
Answer: Magnetic fields add as vectors, just as electric fields do.

superposition principle
B ^→_total = B ^→₁ + B ^→₂
B ^→₁ ↔ magnet 1, and B ^→₂ = ↔ magnet 2.

Magnetic field of two wires

Example: two wires with equal but opposite currents I₁ and I₂ = - I₁
question: is this a parallel or a series circuit?
What would be the force on a third wire?
Determined by the total magnetic field of the two existing wires combined (vector addition):
B ^→_total = B ^→₁ + B ^→₂

Electromagnets

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Build a larger magnetic field by combining many wires with current in same direction
Solution: coiled wires (= "solenoids", from Greek for pipe)
Experiment with solenoids
Magnetic field of a solenoid with N turns of wire:
B = μ₀ I ‎N‏‎L‏
Here, L = length of solenoid, I = current.

Electromagnets

Example: Buzzer
Coil's magnetic field enhanced by an iron core
Magnetic field of a solenoid with additional core:
B = μ I ‎N‏‎ L‏
Here, μ = permeability of the core; can be much larger than μ₀ = 4 π × 10^‎−7‏ (permeability of free space)

The Earth's magnetic field

Earth is not a bar magnet!
The Core is too hot to sustain permanent magnetism.
Magnetic field created by currents of iron-rich molten alloy in the outer core (between solid mantle and Moon-sized inner core), under convection and Coriolis forces = "geo-dynamo"
Magnetic-field reversals on average every 200,000 years

Ferromagnets

How do bar magnets retain their magnetization?
Ferromagnetism:
The atoms themsevles are little magnets, and can align parallel to each other
Unmagnetized: randomly oriented domains
Fully magnetized: atomic magnets in all domains parallel
Examples: Iron; strongest to date: "Neodymium magnets" (GM, 1980's): NdFeB

Faraday's Law of Induction

A bar magnet is moved into and out of a closed wire loop
No battery or other power source!
Measure current I through the loop
Observation:
1. Magnet held still (outside) ⇒ I = 0
2. Magnet inserted ⇒ current flows
3. Magnet held still (inside) ⇒ I = 0
4. Magnet retracted ⇒ current flows in opposite direction

Lenz' rule

Magnetic fields oppose change
Copper ring cooled to ∼70 K
Bar magnet falling through clear, straight tube
Observation:
1. Magnet slows down as it approaches ring
2. Magnet "reluctantly" speeds up again after passing through hole