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Mathematics for Physics

62 Integral Form

An integral statement of Maxwell’s equations that is used to relate fluxes and circulations of the fields to enclosed charge and current, where a flux is the amount of a field that crosses a surface, and where a circulation is the line integral of a field around a loop.

The integral form of Gauss’s law. The net outward electric flux through a closed surface is proportional to the charge enclosed by that surface. A closed surface is a boundary that separates an inside from an outside. This principle is used to compute electric fields of highly symmetric charge distributions.

The integral form of Gauss’s law is

\[ \oint\mathbf{E}\cdot d\mathbf{a} = \dfrac{Q_{\mathrm{enc}}}{\epsilon_{0}} \]

where

  • \(\mathbf{E}\) is the electric field.
  • \(d\mathbf{a}\) is an outward area element of the closed surface.
  • \(Q_{\mathrm{enc}}\) is the enclosed charge.
  • \(\epsilon_{0}\) is the permittivity of free space.

Gauss’s law for magnetism. The net outward magnetic flux through every closed surface is zero. This principle is used to require that magnetic field lines form closed loops.

The integral form of Gauss’s law for magnetism is

\[ \oint\mathbf{B}\cdot d\mathbf{a} = 0 \]

where

  • \(\mathbf{B}\) is the magnetic field.
  • \(d\mathbf{a}\) is an outward area element of the closed surface.

Faraday’s law. The circulation of the electric field around a closed loop equals the negative rate of change of magnetic flux through any surface bounded by that loop. This principle is used to compute induced voltages in generators and transformers.

The integral form of Faraday’s law is

\[ \oint\mathbf{E}\cdot d\mathbf{l} = -\dfrac{d\Phi_{B}}{dt} \]

where

  • \(\mathbf{E}\) is the electric field.
  • \(d\mathbf{l}\) is a directed element of the loop.
  • \(\Phi_{B}\) is the magnetic flux.
  • \(t\) is time.

The Ampère-Maxwell law. The circulation of the magnetic field around a closed loop is proportional to the enclosed current plus the rate of change of electric flux through a surface bounded by the loop. This principle is used to compute magnetic fields of currents and of changing electric fields.

The integral form of the Ampère-Maxwell law is

\[ \oint\mathbf{B}\cdot d\mathbf{l} = \mu_{0}I_{\mathrm{enc}} + \mu_{0}\epsilon_{0}\dfrac{d\Phi_{E}}{dt} \]

where

  • \(\mathbf{B}\) is the magnetic field.
  • \(d\mathbf{l}\) is a directed element of the loop.
  • \(\mu_{0}\) is the permeability of free space.
  • \(I_{\mathrm{enc}}\) is the enclosed current.
  • \(\epsilon_{0}\) is the permittivity of free space.
  • \(\Phi_{E}\) is the electric flux.
  • \(t\) is time.

Note: Also called the integral form of Maxwell’s equations.

62.1 References

  1. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. §2.2.1, §5.3.2, §7.2.1, §7.3.3 — integral Maxwell equations.
  2. Knight, R. D. Physics for Scientists and Engineers. Pearson, 2023. — integral Gauss, Faraday, Ampère laws.
  3. Susskind, L., & Friedman, A. Special Relativity and Classical Field Theory. Basic Books, 2017. — integral versus differential form.