Why isn't the magnetic flux of a loop infinite?

In summary, the flux through a wire loop is very dependent on the thickness of the wire. It is not right to assume that the flux is infinite for a wire loop. The line integral for x-2 diverges near zero, was I wrong in assuming the surface integral (I guess for [r0 - r]-2) similarly diverged?
  • #1
Arijun
21
1
If B~1/r2, then if we have a simple loop, B near the inner edge of the loop will be infinite (or close to it). Why then, would our flux not be infinite?

I also get infinity if I take
∫ ∇ X A *da =∫B*da =∫(closed)A*dl
Since A ~1/r and r~0 at the limit of our surface integral.

I know I am missing something simple here, help! If possible, show where I'm wrong in both of my attempts.
 
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  • #2
You're taking the wire carrying the current as having no thickness. If you consider the wire as nested cylindrical shells, you'll find (e.g. using Ampère's law) that each of these contributes a B outside itself but not inside. Therefore as you approach the wire from outside it, and then enter the wire, the field does not go to infinity.
 
  • #3
That would seem to indicate that flux through a wire loop is very dependent on the thickness of the wire. That doesn't seem right to me. Furthermore, I would be able to make a similar argument for a solenoid for which (to any reasonable approximation) wire size makes no difference. Are you sure that's the only factor I'm missing?
 
  • #4
Your paradox is more general than just the flux case. It is about integration. As you put it, in distances very close to the wire, the field density is very height ( not really infinite), but when you integrate for the flux, you also multiply the high value by an infinitesimal surface. The integral is not necessarily large then.
 
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  • #5
The line integral for x-2 diverges near zero, was I wrong in assuming the surface integral (I guess for [r0 - r]-2) similarly diverged?
 
  • #6
I haven't calculated the flux for the loop, but I have considered a simpler case; the flux produced by a long straight wire (length L) inside the wire itself (radius a) and in the vacuum between it and a cylindrical metal sheath (radius b) acting as return wire. I find, using elementary maths, that
[tex]\Phi = \frac{\mu_0 I L}{2\pi} ln\frac{b}{0.61a}[/tex]

So, in this case, the flux is indeed crucially dependent on the radius of the wire. [The factor of 0.61 is due to the flux inside the wire, and assumes a uniform current density over the wire's cross-section, so no skin effect, for example.]

Some quick thoughts about the solenoid... The local B on the inside wall of the solenoid due just to the adjacent wire is of the order of [itex] B_{loc} =\frac{\mu_0 I}{2\pi r_{wire}}[/itex], whereas the general field in the solenoid is [itex] B_{gen} =\frac{\mu_0 N I}{L_{sol}}[/itex].

But for a single layer coil [itex] N2r_{wire}=L[/itex].

Thus [itex]\frac{B_{loc}}{B_{gen}}= \frac{1}{\pi}[/itex]

suggesting that perturbations to B near the solenoid wall due to the discreteness of the turns
of wire won't be very significant.

[This just an order of magnitude estimate. I'd be the first to acknowledge the crudeness of this treatment.]
 
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  • #7
Arijun said:
The line integral for x-2 diverges near zero, was I wrong in assuming the surface integral (I guess for [r0 - r]-2) similarly diverged?

To my knowledge, close to a thin wire, B is proportional to 1/r and A to ln(r) ( because [itex]B=\nabla \times A =\frac{\partial A}{\partial r}[/itex]) . As we move toward the center of the loop, the dependence become less . But now I think their integrals are till infinite as you said.

I found in Wikipedia that the inductance of a circular coil depends on ln(r/a) where r is the loop radius and " a" is wire radius. So as "a" goes to zero, the inductance becomes infinite which, for a finite current, makes the flux infinite too.
 
  • #8
Yes, the integrals are infinite, but only because you're letting a go to zero (which I think is an odd thing to do)!
 
  • #9
This is actually very interesting ( and odd!). I thought thicker power transmission lines have a higher inductance than thinner ones.
 

1. Why can't the magnetic flux of a loop be infinite?

The magnetic flux of a loop cannot be infinite because it is limited by the strength of the magnetic field and the size of the loop. As the loop gets larger, the magnetic field gets weaker, and vice versa. This creates a natural limit to the amount of magnetic flux that can pass through the loop.

2. How does the size of the loop affect the magnetic flux?

The size of the loop directly affects the magnetic flux. As the loop gets larger, the amount of magnetic flux passing through it increases. However, there is a limit to this increase as the strength of the magnetic field decreases with distance, resulting in a maximum amount of flux that can pass through the loop.

3. Can the magnetic flux of a loop change over time?

Yes, the magnetic flux of a loop can change over time. This can occur when there is a change in the magnetic field strength or a change in the size or orientation of the loop. The change in magnetic flux can also induce an electric current in the loop, according to Faraday's law of induction.

4. Is the magnetic flux of a loop affected by the material of the loop?

Yes, the material of the loop can affect the magnetic flux. Some materials, such as iron, can become magnetized and increase the strength of the magnetic field passing through the loop. This can result in an increase in the magnetic flux through the loop.

5. Why is the concept of magnetic flux important in science?

Magnetic flux is an important concept in science because it helps us understand the relationship between magnetic fields and electric currents. It also plays a crucial role in many applications, such as electric motors, transformers, and generators. Understanding magnetic flux allows scientists to design and optimize these technologies for various purposes.

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