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Physics LibreTexts

15.7: Maxwell's Fourth Equation

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.

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