A curious question on the inapplicability of Faraday's law

[Nanyang]

I recently came across this article on wikipedia : http://en.wikipedia.org/wiki/Faraday_paradox#Inapplicability_of_Faraday.27s_law"

In this example, the circuit does not move, and the magnetic flux through the circuit is not changing, so Faraday's law suggests no current flows. However, the Lorentz force law suggests a current does flow.

I don't understand why is it that Faraday's law is inapplicable here. But wouldn't there be a increase in magnetic flux, since the open surface will increase in area being exposed to the magnetic field. For example, if you have a circular ring of charges and you let the charges move radially outwards, then the area enclosed by this ring will increase.

So if I have a closed loop that first goes in the conducting part of the material in the direction of v X B and then through some other path to complete the loop. Then when the charges move in the direction of v, the closed loop will increase the area of the open surface exposed to B and therefore the flux increases and the current produced is exactly that as predicted using the Lorentz force law. So both laws work well... am I correct?

 

[Born2bwire]

No. The electromotive force in Faraday's law is due to a change in the magnetic flux in time. The magnetic flux is the total magnetic field normal to a surface.

$$\mathcal{E} = - \frac{d}{dt} \int \mathbf{B}\cdot d\mathbf{S}$$

The surface exposed to the magnetic field that we will be taking the flux over is constant for a given time T. This is the time between the left edge of the translating sheet just leaving the "light area" and until the right edge of the translating sheet engers the "light area." During the time that the area illuminated by the light is constant, the flux is constant since the magnetic field is a constant field applied to the same area as the light. This area does not change nor does the magnetic field, hence its time invariance. However, the movement of the sheet itself gives rise to a velocity on the electrons which must experience a Lorentz force from the magnetic field.

A better way to think of it is to assume that the translating sheet is infinitely long. So there isn't a time when the sheet enter or leaves area illuminated by the light and magnetic field.