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Faraday's Law

You should now be familiar with the long integral mathematical form of FARADAY'S LAW of MAGNETIC INDUCTION: in SI units,

$$\oint_{\cal C} \, \Vec{E} \cdot d\Vec{\ell} \; = \; - {\par . . . . . . over \partial t} \SurfInt_{\cal S} \, \Vec{B} \cdot d\Vec{S}$$ (22.5)

where the line integral of $\Vec{E}$ around the closed loop ${\cal C}$ is (by definition) the induced ${\cal{EMF}}$ around the loop and the right hand side refers to the rate of change of the magnetic flux through the area ${\cal S}$ bounded by that closed loop.

  

Figure: Another infinitesimal volume of space.

To make this easy to visualize, let's again draw an infinitesimal rectangular box with the z axis along the direction of the magnetic field, which can be considered more or less uniform over such a small region. Then the flux through the ``Faraday loop'' is just $B_z \, dx \, dy$ and the line integral of the electric field is

$$E_x(y) \, dx + E_y(x+dx) \, dy - E_x(y+dy) \, dx - E_y(x) \, dy .$$

(Yes it is. Study the diagram!) Here, as before, E_y(x+dx) denotes the magnitude of the y component of $\Vec{E}$ along the front edge of the box, and so on. As before, we note that $[E_y(x+dx) - E_y(x)] = (\dbyd{E_y}{x}) \, dx$ and $[E_x(y+dy) - E_x(y)] = (\dbyd{E_x}{y}) \, dy$ so that FARADAY'S LAW reads

$$\left( \DbyD{E_y}{x} \, dx \right) \, dy \; - \; \left( \Db . . . . . . ht) \, dx \; = \; - \left( \DbyD{B_z}{t} \right) \; dx \, dy$$

which reduces to the local relationship

$$\left( \DbyD{E_y}{x} \; - \; \DbyD{E_x}{y} \right) \; = \; - \left( \DbyD{B_z}{t} \right)$$

between the ``swirlyness'' of the spatial dependence of the electric field and the rate of change of the magnetic field with time.

If you have studied the definition of the CURL of a vector field, you may recognize the left-hand side of the last equation as the z component of

$$\begin{eqnarray*}\hbox{\bf curl} \, \Vec{E} &\equiv& \Curl{E} \cr &\equiv& \Hat . . . . . . over \partial x} - {\partial{E}_x \over \partial y} \right) . \end{eqnarray*}$$

The x and y components of $\hbox{\bf curl} \, \Vec{E}$ are related to the corresponding components of $\dbyd{\Vec{B}}{t}$ in exactly the same way, allowing us to write FARADAY'S LAW in a differential form which describes part of the behaviour of electric and magnetic fields at every point in space:

$$\hbox{\fbox{ ${\displaystyle \Curl{E} \; = \; - {\partial \Vec{B} \over \partial t} }$\space } }$$ (22.6)

This says, in essence, that any change in the magnetic field with time induces an electric field perpendicular to the changing magnetic field. Hold that thought.