Induced Electric Fields in Changing Magnetic Fields: Equations and Applications

[pardesi]

suppose there is a region S through which the magnetic field changes with time then is the electric field induced there only or everywhere in space .
if everywhere what equations does it satisfy

 

[Crosson]

Interesting question, the relevant equation is Faraday's law (of induction) and the effects propagate outward from the event at the speed of light, eventually (in classical theory) making a contribution to every point in space.

 

[pardesi]

but going by what maxwell says
$$\nabla \cross E= -\frac{\delta B}{\delta t}$$
if at a point B does not change then no curl of E hence most probably no E .

 

[pardesi]

in fact if we have the surrounding region of E also curl 0 with the condition that
$$\nabla.E=0$$ and the boundary condition we have indeed $$E \equiv 0$$

 

[Crosson]

Sorry I am out of practice with TeX

E = -grad(V) + A'

where A is the vector potential and prime denotes derivative with time. Look for a discussion of retarded potentials to see how A propagates with time, but suffice to say it does, and whatever changes happen to A at a particular point d distance away from me will in principle be felt after a duration t = c/d, where c is the speed of light.

Edit: If B changes at a single point, then that change will eventually propagate away, but your argument assumes that B can change at one isolated point only, when this is impossible under the assumption of continuous fields i.e. the differential form of Maxwell would not apply at that point of discontinuity. Granted, discontinuities can occur in the theory of Maxwell's equations n matter, but I am not qualified to give an answer that takes matter into account.

 

[pardesi]

so no matter where in space the magnetic field changes every point in space feels the change that is there exists a time t when it experiences the field

 

[pardesi]

well the actual doubt arose from a very well known question having a strange field
the question
a bicycle wheel has linearcharge $$\lambda$$ glued to it in the rim which is very thin.Also the spokes ar non conducting.Radius of the wheel is $$b$$.A magnetic field $$\vec B_{0}$$ exists in the circular region around the centre of wheel within radius $$a <b$$ coming out of the plane of the wheel.The field is insytantaneously switched off find the final angular speed of the wheel.

Well the proof started by assuming the flux change across the rim obeys farady's law but doesn't that seem strange .Field actually changes at a point and effects are felt elsewhere

 

[Crosson]

A magnetic field $$\vec B_{0}$$ exists in the circular region around the centre of wheel within radius $$a <b$$ coming out of the plane of the wheel.

The magnetic field exist in the prescribed region, but it does not say that this is the only region where a non-zero magnetic field exist, and in particular the field could not be discontinuous so it must be non-zero around the fringes of the described region.

I am not totally satisfied with the answer I have given concerning discontinuities, but the one about propagation is more solid. If B changes at a point, this induces the curl of E at that point, but this non-zero E acts as a displacement current to produce a B in an (infinitesimal but slightly larger) Amperian loop, which is changing and so produces E in a slightly larger infinitesimal loop etc.