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Reviewing Faraday's law of an induced electric field due to a changing magnetic field

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

In integral form via application of Stokes theorem:

$$ \oint_\Gamma E \cdot ds = \int_s(\nabla\times E) da = -\int \frac{\partial B}{\partial t} da$$

and $\Gamma$ is the closed curve, with $S$ being the surface bounded by it.

My issue is considering changes of those boundaries.

I'll use a simple example to showcase my issue:

A rectangular loop is placed in a magnetic field produced by a magnet enter image description here

  • After sometime $dt$, the magnet was moved generating a change in $\Phi$:

enter image description here

How would I define $\Gamma$ and $S$ w.r.t to the change? enter image description here

In addition, I can increase the complexity of the problem, as the magnet moves away the rectangular loop would deform to a smaller area leading to a $\Delta A$ would $\Gamma$ and $S$'s application have to change with it?

Change in area A as the magnet is moving

I'd assume $\Gamma$ would change as well, since it's bounded by $S$. All to keep track of the induced electric field $E$ due to the changes in both the magnetic field, and the area of the loop due to deformation.

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Between your first and second diagrams I see no change in $\Gamma$ (the perimeter of the loop) or S (the enclosed area), because the loop is stationary. But there are some rather drastic changes in $\vec B$ within the loop. The case is the province of Faraday's law, and the equations you have quoted.

In going from your second to third diagram, the loop shrinks. This takes you out of the realm of Faraday's Law alone. The shrinking loop cuts magnetic flux and induces an emf due to magnetic Lorentz forces on the charge carriers in it. There may also be a Faraday's law emf if the magnet is still moving and causing changes in $\vec B$ over some or all of the loop area.

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  • $\begingroup$ While the loop shrinks, would the boundary $\Gamma$ change with it? To analyze $\Delta A$ as well as $\Delta B$? $\endgroup$ – Geodesic Mar 16 at 15:59
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    $\begingroup$ Yes. You'd evaluate the Faraday's law emf at time $t$ by integrating $\frac{\partial \vec B}{\partial t}$ over the area bounded by the loop at time $t$. You'd add to this the emf due to the boundary moving, taking the field near the boundary as having its value at time $t$. [This reminds me slightly of the formula for the differential a product.] A good thing for you to do would be to set up a simple case to do as an example. $\endgroup$ – Philip Wood Mar 16 at 17:19

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