Terms

• magnetic dipole moment ($\boldsymbol{\mu}$) measures strength of magnetic field and orientation from south to north pole

Creation of Magnetic Fields

Electromagnets

• from a stationary reference frame, the density of electrons and protons in a wire is equal, and the wire is neutral
• from a reference frame moving with the electrons, the protons move backwards, and length contraction causes their spacing to shrink; this increases their density, causing a net force away from the wire for a positive charge
  • for the moving reference frame/positive charge, this is the electric field
  • for the stationary reference frame, the behavior on this moving positive charge is the magnetic field
  • magnetic and electric fields are the same thing, viewed from different reference frames

Permanent Magnets

• electrons have an “intrinsic property” of spin angular momentum or spin
  • since electrons are negatively charged, this “spinning” is not (but can be analogously thought of) as motion
  • this creates a magnetic dipole moment for the electron; the true reason behind this is more known than this explanation, but out-of-scope currently
  • if electrons are unpaired in valence shell, the material acquires a net magnetic moment
• quantum exchange interaction
  • if electrons occupy the same orbitals, their spins must be opposite (according to pauli exclusion principle)
  • for electrons in neighboring orbitals, parallel spin force an antisymmetric wave function which forces the electrons to be far apart (without consuming energy)
  • because the electrons are far apart, there is less Coulomb repulsion; however all the electrons can get in closer to the nucleus, lower energy and stabilizing this configuration
• crystalline pinning
  • an external magnetic field aligns the magnetic domains in a macroscopic magnet
  • in nature, it’s theorized that this is how lightning creates lodestones, the only naturally known permanent magnet

Time-varying Electric Field

• we’ll get to this later in another Maxwell equation

Integral Form of Gauss’s Law for Magnetic Fields

Lorentz Equation $$\mathbf{F} = q\mathbf{v} \times \mathbf{B}$$

Total magnetic flux passing through closed surface is zero $$\Phi_B = \oint_S \mathbf{B} \cdot d\mathbf{A} = 0$$

• because of relativity, the magnetic force is perpendicular to the direction of the moving charge
• this means acquiring the force requires taking a cross product of the magnetic field and moving charge, because cross product is the only perpendicular operator
• for a wire, this means the magnetic field is in loops, with no beginning and end
• therefore, there is no “source” or “sink” of a magnetic field, and all magnets are dipoles
• geometrically, this means all field lines that enter a volume must also exit –> zero flux

Units

• depends on speed of charge, so it’s same as electric field divided by speed’s units: $\frac{\text{N/C}}{\text{m/s}}$

Differential Form of Gauss’s Law for Magnetic Fields

The divergence of the magnetic field at any point is zero $$\nabla \cdot \mathbf{B} = 0$$

• there are no sources or sinks, so the flow into all points equals the flow out of all points