Faraday's Law

From HvWiki

Faraday's Law can be stated in words as:

electromotive force around any closed loop is induced by changes in magnetic flux through the loop.

In integral form:

$\oint_{\partial S} \textbf{E}\cdot d\textbf{l} = -\frac{d}{dt}\int_S \textbf{B}\cdot d\textbf{S}$

Note that the time derivative is the total time derivative, i.e.

$\frac{d}{dt} = \frac{\partial}{\partial t} + \textbf{v}\cdot \nabla,$

where $\textbf{v}\,$ is the velocity of the moving coordinate system where the electric field is observed. By realising that

$\frac{d\textbf{B}}{dt} = \frac{\partial\textbf{B}}{\partial t} + (\textbf{v}\cdot\nabla)\textbf{B} = \frac{\partial\textbf{B}}{\partial t} + \nabla\times(\textbf{B}\times\textbf{v}) + \textbf{v}(\nabla\cdot \textbf{B})$ ,

where $\textbf{v}\,$ is a taken as a constant in the spatial coordinates (consistent with a "rigid" circuit), is much less than the speed of light and that

$\nabla\cdot\textbf{B} = 0\,$ .

Applying Stoke's Theorem to the preceding expression, Faraday's law can be rewritten as

$\oint_{\partial S} \textbf{E}\cdot d\textbf{l} = -\int_S \frac{\partial \textbf{B}}{\partial t}\cdot d\textbf{S} - \oint_{\partial S}(\textbf{B}\times\textbf{v})\cdot d\textbf{l}.$

By assuming that the observer and circuit are in the same inertial coordinate frame, it is relatively straightforward to derive the differential form of Faraday's Law which appears as one of Maxwell's Equations:

$\nabla\times\textbf{E} = -\frac{\partial\textbf{B}}{\partial t}$ .