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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.