How fundamental is Faraday's Law?

[Noesis]

I don't really understand how Faraday's Law can be one of Maxwell's equations if it doesn't seem all that fundamental to me.

An induced emf/voltage is created by the rate of change of magnetic flux with respect to time in a closed loop.

Well obviously if a voltage is induced it means an electric field was created due to this changing magnetic flux. So changing magnetic flux induces an electric field.

Now there is no reason to believe that if we simply 'unbend' a loop that the electric field will cease to be created, so by unbending the loop (there would be zero flux now), and just changing the magnetic field we should create an electric field.

Am I wrong? By this logic a simple changing magnetic field should create an electric field anywhere in space. It would seem that The Law of Electromagnetic Induction would only be an application of this more fundamental law since it has to deal with flux. If I'm on the right page, where the hell is this other law?

If I'm wrong, well then why does this only work for a 'loop'?

 

[jtbell]

You can see the general version of Faraday's law of induction (equation III of the set of four known as Maxwell's equations), written in terms of E and B fields, at

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/maxeq.html

It has two forms, an integral form and a differential form, which you can see on that page, along with the rest of Maxwell's equations. On the left hand side of the integral form, if the path of integration is occupied by a loop of wire, then the integral gives the emf around the loop. But the integral remains true (in terms of the E field) even if there is no wire to "carry" an emf.

They really should write the right-hand side of the equation as an integral too, using the definition of flux:

$$\oint {\vec E \cdot d \vec s} = - \frac {\partial}{\partial t} \int {\vec B \cdot d \vec A}$$

 

[Noesis]

Thank you Jtbell, I found something similar googling of course but I had no real idea what was going on. Your explanation helped.

Please correct me if I'm wrong...my mathematical sophistication is not up to par yet but I am really hoping to understand.

So what is basically happening is that a changing magnetic field creates an electric field, and we can quantify it by looking at the rate of change of the flux that is going through a particular area and realizing that it is proportional to the electric field in the particular path or perimeter of this area?

So if we wanted to find the electric field on a flat sheet of space, or bounded plane, that was created due to a changing magnetic field, we would have to analyze each and every perimeter that is enclosed by the magnetic flux?

We would thus have to integrate the above formula?

Thank you very much in advance Jtbell.

 

[jackiefrost]

Some new insight might be gained from examining the differential form -
curl E = -dB/dt

This just shows that the rate of change in magnetic flux is associated with the existence of an aspect of the electric field, sometimes referred to as "circulation". The circulation of a field is mathematically described by a differential vector operation called "curl". The curl of a vector field (e.g. curl E) describes the field's propensity to produce a rotational effect in whatever it is that the field is defined as influencing.

Richard Feynman had an illustration of circulation, appearing in vol 2 of The Feynman Lectures in Physics. Imagine a thin pool of water. Imagine that the water in this pool is influenced by some force and we can mathematically represent that force by a vector field F. In other words, for each point in space (or, at least the space related to the pool), F assigns to it a vector. That vector describes how much and in what direction the force that F represents, exerts on the water at that point. Also, assume that F is the only influence on the water's motion. Now imagine that immersed in the water is an imaginary circular tube, also containing water. Picture the tube like a nearly invisible hose formed into a a closed loop (i.e. both ends connected) and with zero thickness walls. That is, the tube is just an imagined closed sub-region or volume-loop within the overall volume of the pool's water. Suppose we instantly could freeze the motion of all the water in the pool except that which is found within the tube. If we find the water in the tube possesing some overall flow around the tube, again, with F being the only influence, it would demonstrate an aspect of F called "circulation". Circulation is represented and calulated mathematically by what's called "the curl of F" - see http://mathworld.wolfram.com/Curl.html" [Broken].

When mathematically dealing with rotational phenomena it's common to find vector cross products since implicit in the definition of the cross product is its ability to represent not only the magnitude of rotational influence, but the direction as well. So, not surprisingly, the "curl of F" is a cross product. It's a cross product of a vector function F (or "vector field")with a differential operator called del and the result of this cross-product is another vector function. This function, like the original function F, takes as input a set of coordinates (a point in space) and spits out a vector. However, this vector gives us the direction of the axis of rotation and the magnitude of the rotation that F will exert on a unit of whatever "stuff" F is defined as influencing, at the point F "crunched". This new function can do this for all the points for which the original function F is defined. So, for the pool, if we had an infinite set of infinitessimaly tiny adjacent tubes, the curl of F would tell us the rotational influence that F would have, for each point in the pool, on the tube of water surrounding that point.

If we now relate this concept of circulation as described by the curl of a vector field, to the electric field that accompanys a changing magnetic flux, it may add a different flavor, maybe even some spice, to the peculiar nature of this phenomena.