Magnetic flux rule for calculating motional EMF

[Pushoam]

The above thing is the image.

Please click it.

 

[rude man]

The above thing is the image.

Please click it.

The problem includes the statement " ... I never specified the particular surface to be used ... ". Please include the statement referred to.

 

[Pushoam]

I attached the whole section of the book from which the question is asked.The definition of flux is given in page no. 295, first attachment.

 

[rude man]

I attached the whole section of the book from which the question is asked.The definition of flux is given in page no. 295, first attachment.

OK, thanks. I know the Stokes theorem is appropriate here, but I haven't decided yet on whether the way you invoked the Divergence theorem isn't also OK. That's my answer for the moment & I'll try to get back a bit later with any new views.

 

[Pushoam]

I know the Stokes theorem is appropriate here,

How can one apply stokes' theorem to answer this question ?
Will you please give me some hint?

 

[rude man]

How can one apply stokes' theorem to answer this question ?
Will you please give me some hint?

After looking at your pdf files I see that neither theorem is appropriate for proving that the shape of the surface enscribed by a contour (a loop) is immaterial. I have to apologize to you for going in that direction for so long but i did need to see those pdf pages.

In fact, either theorem requires the inclusion of a maxwell relation. And here's the problem with that: when dealing with moving media such as the loop of fig. 7.13 the maxwell relations are often irrelevant! The author himself points that out (p. 298 lines 8 and 9).

So, bottom line, I conclude that neither the Stokes nor the Divergence theorem is apposite to proving what he seems to be referring to. Referring again to fig. 7.13, the emf is generated differentially for every segment of the loop dl, so the attached surface is immaterial. The loop of fig. 7.13 is an example of where what I call the "Blv law" is the correct law to invoke, not any of the four maxwell relations: d(emf) = B⋅(dl x v) = (v x B)⋅dl. And so the total emf around the loop is just ∫(v x B)⋅dl. The shape of the surface has nothing to do with this integral!

As an example of where you luck out with maxwell is fig. 7.16. In this case emf = - dΦ/dt (based on maxwell's ∇ x E = - ∂B/∂t plus Stokes) happens to be correct but safer is to use the BLv law: emf = Blv based on the Lorentz law F = qv x B.