My year 12 physics textbook teaches the concept of electromagnetic induction in a rather unintuitive way. Perhaps it is not being thorough. Perhaps I am not being thorough in reading it. Here is my problem:
The strength of a magnetic field has so far been described by the Magnetic Flux Density
B: the force experienced by a length of current running perpendicular to the field lines. A few right hand rules make this concept grokable.
Before proceeding to teach induction, Magnetic Flux is defined as the Magnetic Flux Density multiplied by the area of effect:
(|) = B*A. This is already muddying the water: We learn that, in fact,
B is an odd kind of density, in that it is measured within an area, not a volume. The word 'Density' usually describes an amount per volume. Even so, the concept of Magnetic Flux is still not hugely difficult to understand.
The real problem arises when one starts to speak about induction of current in a loop of wire. "The induced current in a conducting loop is proportional to the rate of change of flux". The difficulty is that the amount of flux affecting the wire loop depends on neither the thickness nor (directly) the length of the wire in question, but the area enclosed by the wire: The flux used to decide how much current is induced is that the wire wraps around, rather than some other quantity measured through the region of space that the wire also occupies.
It seems odd to me that the space that the wire does not pass through has anything to do with the current induced in the wire. I present a couple of thought experiments that cause me trouble:
The calculation of induced current apparently only works on closed loops: What of half loops? What of straight wires? They cannot be prescribed an "area" yet I am sure that they can induct.
Imagine a bar magnet attached to a dart. The dart is thrown through loop of wire the size of a watch face, and the current in the loop is recorded. The experiment is repeated, but with a loop the size of a planet. It is not immediately obvious that a wire that is half a world away should induct more current than one in the same room. The magnet in question hardly produces a uniform magnetic field when stationary; is such a uniform field assumed?
Circular wires have greater loop-areas than equivalently long square wires, Yet the calculation is unaffected by the shape of the loop.
The textbook makes no argument as to why the loop's enclosed area is considered. I do not have access to the equipment necessary to test the claims. I am afraid I am missing some bigger picture. So, my question is, why does the area of a loop of wire have anything to do with the induced current (and voltage)? Is there no other, more general, or more intuitive calculation for induction? Perhaps the loop-of-wire calculation is just a special case of this.