Why does induced current depend on the area of a loop of wire?

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:

1. 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.

2. 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?

3. 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.

• It's worth noting that it's the induced voltage, not the induced current that's equal to the rate of change of magnetic flux through the loop. So in the case of your wire loop "the size of a planet", the longer wire would have much more resistance than the wire loop "the size of a watch face"; and you'd end up with much less current as a result. Aug 2 '16 at 1:25
• Yes, sorry, I what the book says is that current and change of flux are proportional. Aug 2 '16 at 1:40
• OK, another thought - and maybe I should start to assemble these comments into an answer. If you're going to induce a current, there needs to be a loop for the current to flow in. You're not going to get current flowing along a straight wire or a half loop, unless the two ends are joined up. And as soon as you join the ends, you've got an area in which you can measure the rate of change of flux. Aug 2 '16 at 1:51
• And if the loop is bent? like the boundary of a pringle? What then? The whole loop concept still seems a little awkward. The area would then be that of the shadow of a pringle, right? Also, a voltage can exist without flow - like in a capacitor. Aug 2 '16 at 1:53
• I don't see what difference it makes if the loop is bent. Imagine if a soap bubble were to form on the loop. You could imagine the flux density at each point on the soap bubble; and (roughly speaking) add them all up. Aug 2 '16 at 1:57

Imagine a wire shaped like a long thin rectangle, with an ammeter at one end. It's in a constant magnetic field that's perpendicular to the plane of the rectangle. Now move the rectangle in a direction that's perpendicular to the two long sides of the rectangle, and also to the magnetic field; just like the blue arrow in this diagram. This motion pushes the electrons along the top side in one direction, and the electrons along the bottom side in the same direction. In terms of current flowing in the loop, the two pushes cancel each other out, and the ammeter reads zero.

Now try the same experiment in a magnetic field that's not constant. Let's say the field is stronger towards the top of the diagram, and weaker towards the bottom. This means that the push of the electrons along the top side of the loop will be stronger than the push of the electrons along the bottom side of the loop, and they won't cancel each other out exactly. A net current will flow, and the ammeter will show the current, while the wire is moving.

But what you've also done is move the loop into a region where the field strength is stronger. While you were moving the wire, the total magnetic flux in the loop was increasing, because the top part of the wire passed through a stronger magnetic field than the bottom part did. Combining Faraday's law and Ohm's law tells us that the current induced is proportional to the rate of change in magnetic flux. $$I = - \frac {d \iint B \cdot dA}{R dt}$$ Here, "magnetic flux" is basically a total of the magnetic field strength at every point within the loop. (Technically, this total is a surface integral, which is like adding up a quantity across every little piece of the area).

You can see that it makes no difference at all what shape the loop is - what matters is that there's a difference in field strength between one part of the loop and the other.

And yes, there has to be a loop, because current can only flow in a loop. Even if part of the loop was a straight line, or a half circle, you'd still have to join up the ends to get the current to flow.

Now, in answer to your point about the bar magnet attached to the dart. The magnetic field around a bar magnet passes through the magnet itself, from South to North; and then outside the magnet from North to South. So if you have a huge loop around the magnet, part of the loop will contain a magnetic field in one direction, and part will contain a magnetic field in the opposite direction; and the net magnetic flux will be very close to zero. Firing your bar magnet through a huge loop won't create much change in flux, so it won't induce much current. But a very small loop might enclose the field that passes inside the magnet, but not very much of the field that passes outside the magnet. If you pass your bar magnet right through the small loop, this might cause a current to be induced, first in one direction as the magnet enters the loop, and then in the other direction as the magnet exits the loop.

That is a good question. The flux lines add up in the central area (and cancel outside) and so it is this changing flux (in this area) that has the largest effect. Try drawing the field lines around the individual wire on paper. By Lenz's law, there is an opposing flux (to the inducing flux) - this is also most concentrated in central area. I.e. Most of the induction action happens in the central area. - Sebula R