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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.