Skip to main content
\(\require{cancel}\)
Physics LibreTexts

15.7: Maxwell's Fourth Equation

  • Page ID
    5342
  • [ "article:topic", "Lenz\'s law", "Maxwell\u2019s equations", "Faraday\u2019s Law", "authorname:tatumj" ]

    This is derived from the laws of electromagnetic induction.

    Faraday's and Lenz's laws of electromagnetic induction tell us that the E.M.F. induced in a closed circuit is equal to minus the rate of change of B-flux through the circuit. The E.M.F. around a closed circuit is the line integral of \(\textbf{E} \cdot \textbf{ds}\) around the circuit, where \(\textbf{E}\) is the electric field. The line integral of \(\textbf{E}\) around the closed circuit is equal to the surface integral of its curl. The rate of change of B-flux through a circuit is the surface integral of \(\dot{\textbf{B}}\). Therefore

    \[\textbf{curl}\, \textbf{E} = - \dot{ \textbf{B}} \tag{15.7.1} \label{15.7.1}\]

    or, in the nabla notation,

    \[\boldsymbol{\nabla} \times \textbf{E} = - \dot{ \textbf{B}}. \tag{15.7.2} \label{15.7.2}\]

    This is the fourth of Maxwell's equations.

    Contributor