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.

 

[astrokreng]

Thank you for bringing this interesting topic to my attention. I can understand your confusion regarding the inapplicability of Faraday's law in this scenario. However, it is important to note that Faraday's law is based on the principle of electromagnetic induction, which states that a changing magnetic field can induce an electric current in a conductor.

In the example you provided, the circuit is not moving and the magnetic flux through it is not changing. Therefore, according to Faraday's law, no current should be induced. However, the Lorentz force law takes into account the motion of charges and their interaction with a magnetic field, which can result in a current being induced even in the absence of a changing magnetic flux.

In the case of the circular ring of charges, the area enclosed by the ring does increase as the charges move radially outwards. This does result in an increase in the magnetic flux through the loop, which in turn induces a current according to Faraday's law. However, in the example given in the article, the circuit is not moving and the magnetic flux through it is not changing, making Faraday's law inapplicable.

In conclusion, while both Faraday's law and the Lorentz force law may work well in certain scenarios, it is important to understand the principles behind each law and the conditions under which they are applicable. Thank you for your curious question and I hope this explanation helps to clarify any confusion.