Why Does Faraday's Law of Induction Consider Coil Area Instead of Length?

[Galadirith]

Homework Statement

Hi guys, I've recenlty done Faraday's law of induction in my AS physics class. I am really not understanding why it works. I get the Maths that's fine (PS were not doing integrals etc, were only doing the maths which invlove say given areas etc), but I don't see why its so.

So the Emf is proportional to the rate of change of Area, now I simply don't get this. I understand if the area is in reference to a surface that has some sort of continuous manifold, but we havnt even done anything like that, in fact the area in question is the area of a coil of wire. This really frustrates me as surly you would not want to be considering the area inside the coil, but rather the length of the coil because the coil wire is the only thing cutting lines of flux. Why is this so? Thanks guys.

Homework Equations

$$\mathcal{E} = -NA\frac{dB}{dt}$$

$$N = Number\ of\ coils\ in\ wire$$

$$A = Area\ inside\ coil$$

$$B = Magnetic \Flux\ Density$$

 

[Defennder]

So the Emf is proportional to the rate of change of Area, now I simply don't get this. I understand if the area is in reference to a surface that has some sort of continuous manifold, but we havnt even done anything like that, in fact the area in question is the area of a coil of wire. This really frustrates me as surly you would not want to be considering the area inside the coil, but rather the length of the coil because the coil wire is the only thing cutting lines of flux. Why is this so? Thanks guys.

You need a closed conducting loop for current to flow in the first place, and Faraday's law says that the induced emf is given by the rate of change of magnetic flux. In this case you're varying area, so it makes sense to define the area as that enclosed by a conducting loop since the emf will be induced around the loop. And it's not true that we're not considering the length of the loop at all. The length of the loop affects N, the number of coils, so that one is already factored into consideration.

 

[Galadirith]

Thanks Defennder, however I still don't see why that should be so. If we just consider that there is a single loop so N = 1. How then are we considering the length of the wire. The only Physicsical thing that is changing (if we only change the length) is the length of the wire. Now I would imagine that the the induced emf would be proportional to the length of wire, but instead is proportional to the Area enclosed by the wire, so the length (or really the circumference) of the single coil is in a quadratic proportion with the induced emf. You said it makes sense to define the area as that is what you are varying. To me that makes no sense, you are varying the circumference primarily and then as a result of that you would be varying the area. Is there a physics theory that describes why, in fact Faraday probably describes it somewhere which i havnt found. I just can't accept that the area is varying so that's on of the variables, there has to be a physical reason why the area should be on of the variables. Thanks

I should have added in my first post:
$$\mathcal{E} = -NB\frac{dA}{dt}$$

 

[Defennder]

Your equation says it all. You have dA/dt. The others N,B do not vary in this setup. The coil is cutting the magnetic flux lines, that itself is an indication the area and hence magnetic flux through some closed loop changing. It is not so much the fact that it is cutting the flux lines than the fact that cutting the flux lines implies that the flux through the loop is changing and that, by Faraday's law induces emf.

If it is the case that cutting flux lines would do, then consider taking a rod (not a closed loop) and moving it across the flux lines of a constant B field. Would you get an emf and voltage generated across the ends by doing so?

 

[tiny-tim]

So the Emf is proportional to the rate of change of Area …

Hi Galadirith!

Nooo … the emf is proportional to the rate of change of flux.

In this case, the field, and its flux, are stationary, and the wire is moving, but it's still the change of flux that's important.

In this case, flux is (obviously ) proportional to area, but in most cases the area will be fixed.

 

[Galadirith]

Thanks guys, tiny-tim infact that was gonning be my next progression, and thanks for pointing it out and taken in from def too, But still that is my question why is the equation so, I know the equation does say it all, I understand that fine, but why is that equation so. Oh k mabye I should ask this question in a different way, considering from the perspective of the electrons in the wire coil. The only field interacting with those electrons in the wire is the flux lines that cut them, so when you are changing the area of the coil in whatever manner that may be why should the area be relevant, it is only the electrons in the loop itself that are actually anything to do with the induced emf.

It not just enough to quote from theories, I know and understand the Math. I should ask then is there even a known reason why this is so, I know this is all easily proven experimentally, so is that the only reason why we accept it?

 

[Defennder]

If it is the case that cutting flux lines would do, then consider taking a rod (not a closed loop) and moving it across the flux lines of a constant B field. Would you get an emf and voltage generated across the ends by doing so?

I was thinking about this, and it turns out that perhaps there may be an induced voltage after all for this. After all, electrons in a falling rod would experience a magnetic force which pushes them to one end of the rod, creating a potential difference across the ends.

 

[Galadirith]

Defender I think also there is an induce emf in a rod, I know not exactly the same but similar I suppose to a capacitor where there is a potential difference across the plates, I think what I am really after is some sort of quantum explanation of the equations. I don't think any macroscopic level physicality's could actually describe what is going on. All that has been disused here must all be results of what is going on at a quantum level, any thought? :-)

EDIT: In fact think aout it further, I think that it is only a current that is not induced in a rod, and when the rod is connect so as to form a loop a current is produced as a result of the induced emf, something like that.

 

[tiny-tim]

… I think what I am really after is some sort of quantum explanation of the equations. I don't think any macroscopic level physicality's could actually describe what is going on. All that has been disused here must all be results of what is going on at a quantum level, any thought? :-)

Hi Galadirith!

No … it is macroscopic … it has nothing to do with quantum theory … Faraday died in 1867!

Oh k mabye I should ask this question in a different way, considering from the perspective of the electrons in the wire coil. The only field interacting with those electrons in the wire is the flux lines that cut them, so when you are changing the area of the coil in whatever manner that may be why should the area be relevant, it is only the electrons in the loop itself that are actually anything to do with the induced emf.

The electrons feel the electric field, E.

(a magnetic field, B, of course, has no effect on stationary charge)

The energy given to the electrons is the work done on them, ∫E.dl

From Faraday's law (which is really geometry … in differential geometric language, it says the electromagnetic field, which is a 2-form derived from a 1-form potential, has zero differential), ∫E.dl = -(∂/∂t)∫∫B.darea = -(∂/∂t)∫∫flux(B).

So you're right in thinking that it should be related to the length of wire … it is, with E … but that happens to be geometrically connected with the area, with B

Measuring the component of E all the way along the wire could be a computational nightmare. So instead we try to create reasonably uniform fields, and measure the flux of B instead.

 

[Galadirith]

Thanks so much guys and Tiny-tim thanks so much for that, That is basically exactly what I need I good Mathematical geometrical interpretation. I now need to consider all this in my mind, I can't say I understand that maths fully, I havnt done any PDE's yet still only in sixth form in the UK. But I think I can rest my mind a bit easier now lol thanks :-)

I know a little off topic but could I ask tiny-tim (if you are indeed and Uni or gone to Uni) what course did you take, I am in a real dilema, I love maths, my true passion, but I want it to be applicable, so i don't know wheather to maths of physics. Geomemtry just so happens to be my absolute fav area of Maths, and seeing as you posted about geometry just then I was wondering what you studied, Sorry if that too off topic, Thanks guys.

 

[tiny-tim]

I know a little off topic but could I ask tiny-tim (if you are indeed and Uni or gone to Uni) what course did you take, I am in a real dilema, I love maths, my true passion, but I want it to be applicable, so i don't know wheather to maths of physics. Geomemtry just so happens to be my absolute fav area of Maths, and seeing as you posted about geometry just then I was wondering what you studied, Sorry if that too off topic, Thanks guys.

Hi Galadirith!

If you love :!) geometry (and if you're good at the other math stuff, like calculus), then definitely do maths.

hmm … or mathematical physics …
​

 

[Galadirith]

Thanks tiny-tim, I definatly have real think about that then, if you don't mind me asking what did you do?