Direction of induced electric field?

[PumpkinCougar95]

If there is a very very big(infinitely big) region of space where $\frac {dB} {dt} = constant$ what would be the E field at any point? Obviously $\nabla x E = constant$ but what after that ?

 

[Charles Link]

I don't think I have a complete answer to your question, but a thread recently appeared that might be somewhat helpful. The induced electric field does not require any local $dB/dt$ for $E$ to be non-zero at that point. See https://www.physicsforums.com/threads/flux-through-a-coil.940861/page-2#post-5952718 $\\$ One additional comment, in writing the "curl" in Latex, use "\" and "times" together to get the $\times$ sign.

 

[PumpkinCougar95]

This post has made me wonder even more. Should there be an emf in this case too?
https://drive.google.com/open?id=1tuqI1S7juh8iZhb8t2TaPrVm82CaL5mU
I am saying this because the induced E field is stronger the closer you get to the region with changing magnetic field.

And If it does, Doesn't this kind of violate the fact that the flux through the loop should be changing to create an E field?

 

[Charles Link]

And a couple of other very recent threads where Faraday's law has just appeared are https://www.physicsforums.com/threa...irection-in-faradays-law.941728/#post-5956349 and https://www.physicsforums.com/threads/induced-emf-and-current.941717/#post-5956172

 

[Charles Link]

This post has made me wonder even more. Should there be an emf in this case too?
https://drive.google.com/open?id=1tuqI1S7juh8iZhb8t2TaPrVm82CaL5mU
I am saying this because the induced E field is stronger the closer you get to the region with changing magnetic field.

And If it does, Doesn't this kind of violate the fact that the flux through the loop should be changing to create an E field?

I can't see the image in this post. Maybe it will become visible in a few minutes...

 

[PumpkinCougar95]

loop should be changing to create an E field

I meant inducing a current.

Edit: Here it is https://drive.google.com/open?id=1tuqI1S7juh8iZhb8t2TaPrVm82CaL5mU

 

[Charles Link]

@PumpkinCougar95 I think you also might find Professor Lewin's video in post 52 of this thread very interesting: https://www.physicsforums.com/threa...asure-a-voltage-across-inductor.880100/page-3

 

[PumpkinCougar95]

And If it does, Doesn't this kind of violate the fact that the flux through the loop should be changing to create an E field?

So what do you think?

 

[Charles Link]

So what do you think?

You can have an $E$ field at locations on a loop where there is zero changing magnetic flux through that loop. That is ok. In traveling around that loop, you would find $\oint E \cdot ds =0$. This doesn't mean that $E$ needs to be zero everywhere on the loop for the integral to be zero. (Hopefully this answers your question here). This result follows from $\nabla \times E=-\frac{\partial{B}}{\partial{t}}$, integrated over an area, along with Stokes theorem. The result is quite exact. $\\$ Additional note: In many cases, computing the $E=E(r)$ and performing a path integral of this $E$ over even a simple loop like a circle could be quite difficult. Stokes theorem gives this result immediately. Doing the calculation the long way might take a couple of hours or longer to get this zero result.

 

[PumpkinCougar95]

You can have an EE E field at locations on a loop where there is zero changing magnetic flux through that loop. That is ok. In traveling around that loop, you would find ∮E⋅ds=0∮E⋅ds=0 \oint E \cdot ds =0 . This doesn't mean that EE E needs to be zero everywhere on the loop for the integral to be zero.

But $\oint E \cdot ds =0$ is NOT true in this case, Even though flux is zero through the loop at all times:

https://drive.google.com/open?id=18tjoiyAjjD1XAXgmZn_3UKgbtMrFZ6zz

 

[Charles Link]

But $\oint E \cdot ds =0$ is NOT true in this case, Even though flux is zero through the loop at all times:

https://drive.google.com/open?id=18tjoiyAjjD1XAXgmZn_3UKgbtMrFZ6zz

You have two parts of the loop where $E$ is perpendicular to $ds$ giving zero. Then you have two radii, $\gamma_1$ and $\gamma_2$. $E(r)=(A)(\frac{dB}{dt})/(2 \pi r )$, with $A=\pi R^2$. For one arc, the path length is $L_1= \gamma_1 \theta$. For the other arc, the path length is $L_2=-\gamma_2 \theta$, ( with a minus sign). $E_1=C/\gamma_1$ and $E_2=C/\gamma_2$ for the same constant $C$. Thereby, $E_1 L_1 +E_2 L_2=0$ This one clearly has $\oint E \cdot ds=0$.

 

[PumpkinCougar95]

oh, Thanks a lot! Now, what about the first question?

If there is a very very big(infinitely big) region of space where $\frac {dB} {dt} = constant$ what would be the E field at any point? Obviously $\nabla x E = constant$ but what after that ?

Is it possible to find this out without some sort of boundary condition?

 

[Charles Link]

I don't know that this first question that you have has a simple answer. It might be in the category of the type of question contained in this recent thread on Physics Forums, and I don't know whether they ever reached a conclusive answer to it: https://www.physicsforums.com/threads/breakdown-of-gauss-law.926835/