The Boundary-Value Problem

Electromagnetic Theory of Static and Quasi-Static Fields in 17 chapters — from “The field as a differential equation: Poisson, Laplace and the well-posed problem” to “Magnetic diffusion, shielding and flux exclusion”.

Chapters

  1. The field as a differential equation: Poisson, Laplace and the well-posed problem
    • From sums to Laplace's equation
    • The mean-value property
    • The relaxation method
    • Earnshaw's theorem
    • Dirichlet and Neumann boundary conditions
    • The well-posed boundary-value problem
    • The boundary condition at infinity
    • Problems
  2. Uniqueness, images and the Green function of a region
    • Green's identities and the first uniqueness theorem
    • The second uniqueness theorem
    • The method of images
    • The image force and energy
    • The image charge for a sphere
    • The Dirichlet Green function
    • Failure of the image method
    • Problems
  3. Separation of variables in Cartesian geometry
    • Superposition of boundary problems
    • Separation of variables
    • The separation constant
    • The series solution
    • Coefficients by orthogonality
    • The closed-form sum
    • Convergence at a corner
    • Problems
  4. Separation of variables in spherical geometry
    • Separation in spherical coordinates
    • Legendre polynomials
    • The Legendre series and the regularity condition
    • Legendre coefficients from boundary data
    • Conducting sphere in a uniform field
    • Azimuthally dependent boundary data
    • Problems
  5. The multipole expansion of the potential
    • The Legendre generating function
    • The multipole expansion
    • The quadrupole moment
    • Origin dependence of multipole moments
    • The point dipole field
    • Truncation error and the interior multipole expansion
    • Problems
  6. Cylindrical geometry, edges, and what a field does to a dipole
    • Separation in cylindrical coordinates
    • Two-dimensional multipole series and the cylinder
    • The field near an edge
    • Bessel functions
    • Dipole potential energy
    • Force and torque on a dipole
    • Problems
  7. Polarisation and bound charge, completed
    • The potential of a dipole distribution
    • Bound charge densities
    • The uniformly polarised sphere
    • The average field inside a sphere
    • Interface conditions on the potential
    • The cavity field for three shapes
    • Indeterminacy of the displacement field
    • Problems
  8. Linear dielectrics and the boundary-value problem with matter in it
    • A source-free dielectric interior
    • The dielectric boundary-value problem
    • Dielectric sphere in a uniform field
    • The sphere response factor
    • The dielectric image charge
    • Limits of linearity
    • Problems
  9. Energy and force in dielectrics
    • Work done by the free sources
    • Two energy expressions
    • Force by virtual work
    • Force on a dielectric slab
    • Fixed charge and fixed voltage
    • Problems
  10. Molecular polarisability, permanent dipoles and the Clausius-Mossotti relation
    • Polarisation of two gases
    • Electronic polarisability
    • Orientational polarisation
    • The Debye equation
    • Local field and the Lorentz cavity
    • The Clausius-Mossotti relation
    • The polarisation catastrophe
    • Problems
  11. The vector potential and the magnetostatic boundary-value problem
    • Magnetic flux of a solenoid
    • Vector potential and gauge freedom
    • Coulomb gauge and the vector Poisson equation
    • Vector potential of a solenoid
    • Interface conditions on the vector potential
    • Gauge freedom and its limits
    • Problems
  12. The magnetic multipole expansion and the magnetic scalar potential
    • The Earth's magnetic dipole moment
    • Multipole expansion of the vector potential
    • Magnetic moment of a current distribution
    • Magnetic dipole in an applied field
    • The magnetic scalar potential
    • A multivalued scalar potential
    • Problems
  13. Magnetisation, bound currents and H, completed
    • The vector potential of a magnetised body
    • The uniformly magnetised sphere
    • The demagnetising factor
    • The demagnetising correction to a measurement
    • The linear magnetisable sphere
    • Indeterminacy of the auxiliary field
    • Problems
  14. Diamagnetism, paramagnetism and magnetic susceptibility
    • Diamagnetism and paramagnetism
    • The Larmor response
    • The diamagnetic susceptibility of atoms
    • Langevin paramagnetism and the Curie law
    • Curie plots and paramagnetic saturation
    • Diamagnetic levitation
    • The Bohr-van Leeuwen theorem
    • Problems
  15. Ferromagnetism: the Weiss field, domains, hysteresis and the magnetic circuit
    • The Weiss molecular field
    • The Curie-Weiss law
    • Magnitude of the Weiss field
    • Magnetic domains and anisotropy
    • Work done on a magnetic medium
    • The hysteresis loop and hysteresis loss
    • The magnetic circuit
    • Problems
  16. The quasi-static approximation, eddy currents and the skin effect
    • Hysteresis loss and the laminated core
    • The quasi-static approximation
    • The magnetic diffusion equation
    • The skin depth of a driven conductor
    • Eddy currents in the skin
    • Eddy-current loss in a lamination
    • Frequency scaling of core loss
    • Limits of the quasi-static approximation
    • Problems
  17. Magnetic diffusion, shielding and flux exclusion
    • A falling magnet and eddy currents
    • The diffusion transient in a slab
    • The magnetic diffusion time
    • Eddy-current braking in a copper pipe
    • Magnetic shielding at low and high frequency
    • The perfect conductor limit
    • The magnetic Reynolds number
    • Problems