Maxwell's/Faraday Law Concern Propagation of Induced Fields

[Electric to be]

The integral form of Maxwell's equation pertaining to induced electric fields is:

∮E(t0)⋅dℓ= −dΦ(t)/dt|t=t0Say for a long time, in some circular region there has been no B or E fields present. Then, there is a sudden constant increase of B field introduced in the middle. I know that information cannot propagate faster than the speed of light, but doesn't Maxwell's equation predict that at that instant, there should be some nonzero integral of the E field at some radius R away from the newly introduced B field? Is Maxwell's equation wrong in this area? This equation written explicitly doesn't somehow tell me that, "Oh there will be a field there, but only after a few instants once there has been sufficient time for that information to spread" haha.

Or does the integral form somehow not hold? (I thought integral and differential form are equivalent, by Stoke's/Divergence Theorem)

Thanks for any help in clearing up this doubt.

 

[Student100]

Not sure if I'm reading the question correctly, but...

$$\nabla \times E(x,y,z,t) = -\frac{\partial B(x,y,z,t)}{\partial t}$$

Which says the curl of the electric field at one instant in time at some point is proportional to the time rate of change of the magnetic field at the same point and instant in time.

Does that clear it up? Can you maybe reword if not.

 

[Electric to be]

Not sure if I'm reading the question correctly, but...

$$\nabla \times E(x,y,z,t) = -\frac{\partial B(x,y,z,t)}{\partial t}$$

Which says the curl of the electric field at one instant in time at some point is proportional to the time rate of change of the magnetic field at the same point and instant in time.

Does that clear it up? Can you maybe reword if not.

So this is the differential form of the equation that I wrote. However, I simply provided the integral form and a situation which seems to violate it. (They are identical by Stoke's theorem)

However, I think I figured it out. At least this is my guess. It isn't possible to provide a constant changing flux without also having the electric field. By this I mean even if I try to provide a constant rate of change of flux, it will stay zero until the wave propagates to the radius.

 

[Dale]

Then, there is a sudden constant increase of B field introduced in the middle.

This type of B field is not possible. If you think about flux lines they must all be closed loops. To get a net flux through the surface, one part of the loop must propagate until it crosses the edge. That only happens at the speed of light.