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Mathematics for Physics

234 Fields

A quantity assigned throughout space that is used to describe how an influence is spread through the surroundings.

The assignment of a quantity through space. A field defines a physical quantity at every point of a region. A scalar field assigns a number at each point. A vector field assigns a direction and a strength at each point. This principle is used to replace a list of particle-to-particle forces by a function on space.

Field energy. A field can store energy and momentum in space. This principle is used to localize field energy even where no matter is present.

The electromagnetic energy density is

\[ u = \dfrac{1}{2}\left(\epsilon_{0}E^{2} + \dfrac{1}{\mu_{0}}B^{2}\right) \]

where

  • \(u\) is the energy density.
  • \(E\) is the electric field magnitude.
  • \(B\) is the magnetic field magnitude.
  • \(\epsilon_{0}\) is the permittivity of free space.
  • \(\mu_{0}\) is the permeability of free space.

Local mediation. A field mediates interactions so that influence need not be instantaneous action at a distance. This principle is used to write local field equations that link the field to nearby sources.

Radiation at finite speed. A changing field can propagate as radiation and carry a history of past sources, because influence travels at finite speed. This principle is used to treat light as a free electromagnetic field.

The force on a test particle. The field at a point determines the force on a test particle placed there. A test particle is a small charge or mass used to probe the field. This principle is used to compute acceleration from \(\mathbf{F}=q\mathbf{E}\) or \(\mathbf{F}=m\mathbf{g}\).

The electric force on a test charge is

\[ \mathbf{F} = q\mathbf{E} \]

where

  • \(\mathbf{F}\) is the force.
  • \(q\) is the test charge.
  • \(\mathbf{E}\) is the electric field.

Note: Also called a field. Upon quantization the field behaves as oscillators whose excitations are particles.

234.1 References

  1. Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering. Cambridge University Press, 2006. — continuous assignment through space.
  2. Susskind, L., & Cabannes, A. General Relativity: The Theoretical Minimum. Penguin Books, 2023. — field as a function on space; tensor character.
  3. Feynman, R. P., Leighton, R. B., & Sands, M. The Feynman Lectures on Physics, Vols. I–II. — mediation, radiation, retardation, field lines.
  4. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — energy in fields; force on a test particle.
  5. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — field energy and momentum; local equations.
  6. Schwartz, M. Principles of Electrodynamics. Dover, 1972. — independent dynamical field.
  7. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — scalar, vector, and tensor fields.
  8. Susskind, L., & Friedman, A. Quantum Mechanics: The Theoretical Minimum. Basic Books, 2014. — field oscillators and particles.
  9. Carroll, S. M. Spacetime and Geometry. Cambridge University Press, 2019. — quantized fields.