This is a continuation of my previous question:
This is the concerned quote form Purcell's Electricity and Magnetism's Electromagnetic Induction:
You may think that one of the rings we considered earlier would have made a simpler example to illustrate the calculation of self-inductance. However, if we try to calculate the inductance of a simple circular loop of wire , we encounter a puzzling difficulty . It seems a good idea to simplify the problem by assuming that the wire has zero diameter. But we soon discover that, if finite current flows in a filament of zero diameter, the flux threading a loop made of such a filament is infinite! The reason is that the field $B,$ in the neighbourhood of a filamentary current, varies as $1/r,$ where $r$ is the distance from the filament, and the integral of $B\times \mathcal{area}$ diverges as $\int (dr/r)$ when we extend it down to $r= 0\;.$ To avoid this, we may let the radius of the wire be finite, and not zero, which is more realistic anyway. This may make the calculation a bit more complicated, but that won't worry us. The real difficulty is that different parts of the wire now appear as different circuits, linked by different amounts of flux. We are no longer sure what we mean by the flux through the circuit. In fact, because the electromotive force is different from the different filamentary loops into which the circuit can be divided, some redistribution of current density must occur when rapidly changing currents flow in the ring. Hence the inductance of the circuit may depend somewhat on the rapidity of $I,$ ...
We avoided this embarrassment in the toroidal coil example by ignoring the field in the immediate vicinity of the individual turns of the winding. Most of the flux does not pass through the wire themselves, and whenever that is the case the effect we have just been worrying about will be unimportant.
I didn't understand the difficulty faced as stated by Purcell when a non-zero diameter wire is taken into account. Why did he say " different parts of the wire now appear as different circuits, linked by different amounts of flux"? "We are no longer sure what we mean by the flux through the circuit"- Why is it so? why can't we define flux for the non-zero diameter wire?
Really I'm not getting why different parts of a non-zero diameter wire become different circuits:/
Can anyone please explain this to me why flux can no longer be defined and how different parts of the non-zero diameter become different circuits?
Answer
In science you need to make predictions. To make a prediction you have to predict something will happen before you see it happens. Which means you need to use your mind to figure out what you think will happen.
A common way to do this is to use your mind to make a mathematical model and then manipulate the mathematical model with your mind to come up with your prediction.
So you need to be able to do some mathematics and to know what real world things your mathematics corresponds to.
The kind of mathematics you can learn includes (but is not limited to) scalar fields, vector fields, gradients, divergences, curls, line integrals, area integrals, volume integrals, fluxes through surfaces and so on.
Sometimes a mathematical result will tell you that the line integral around a mathematical closed curve (a loop) is equal to the mathematical flux through a mathematical surface enclosed by the mathematical loop.
So, for instance the line integral of a vector field $\vec E$ around a mathematical loop might be equal to the flux of $\vec \nabla \times \vec E$ through a mathematical surface bounded by that loop. That's all mathematics.
If you tried to relate this mathematical result about 1d mathematical curves to the real world you'd essentially have two options. The first option would be to model your wire as a 1d loop, in which case you have to deal with a diverging magnetic field as you get close to the wire. The second option is to have a thick wire and realize there are many possible 1d curves through that wire, each having a different line integral of the electric field (modelled by a vector field) along one of the many 1d curves through the wire.
Experimentally you might try to relate the line integral of the electric field to an EMF or even a current through a wire. But if different parts of the wire have different electric fields, and there are many curves that loop through the wire. No single one of them has a direct and obvious correspondence to an EMF. And the current might not be as simple as a cross section all having a $\vec J$ piercing through it orthogonally.
If you had an annulus shaped wire instead of a 1d curve, and all the current was going in a circular direction. Then you could compute the EMF around the inner loop or around the outer loop. And you'd get different results. Neither directly applies to the loop as a whole. You could also compute an EMF around a loop with intermediate radius. Or one that wiggles between the inner and outer radius. When the wire was modelled by a 1d mathematical curve, your line integral could be "around the loop" and by staying in the loop, you had exactly one curve. Now, there are lots of curves that stay in the loop.
If your current stays in your loop, a natural direction to do a line integral is the direction the current is going, but there is no guarantee that such a curve is closed. But for the example we looked at, such a curve is closed.
And each circular loop has its own line integral. And for each loop you might also look at the flux of $-\partial \vec B /\partial t$ through any of the mathematical surfaces bounded by each one of the mathematical loops.
So for instance you could make a mathematical model where there is some $\vec J$ going around the wire. And find the magnetic field. And find the flux through every loop (the inner one, the outer one, and the ones in between). And find for each loop the rate the magnetic flux is changing in time, and relate that to the $\vec J$ along that curve.
From Maxwell you might expect that $\vec \nabla \times \vec E=-\partial \vec B /\partial t.$ And mathematically you might expect that the line integral of the electric field around a closed loop equal the flux of the curl through the surface bounded by the loop. So if $\vec J=k\vec E$ (a small scale version of Ohm's law) then we have \begin{align}\oint_{\partial S} \vec J\cdot d\ell &=k\oint_{\partial S} \vec E\cdot d\ell\\&=k\iint_{S} \left(\vec \nabla \times \vec E\right)\cdot \hat n dA\\&=k\iint_{S} \left(-\frac{\partial \vec B}{\partial t} \right)\cdot \hat n dA,\end{align} where $S$ is a surface bounded by $\partial S$ which is one of the loops within the wire.
So lots of loops, lots of different surfaces. But if you make a model with a certain $k$ and a certain magnetic field and $\vec J$ you can confirm that your model matches above.
And the fact that you restricted your models to ones where \begin{align}\iint_{S} \left(\vec \nabla \times \vec E\right)\cdot \hat n dA &=\iint_{S} \left(-\frac{\partial \vec B}{\partial t} \right)\cdot \hat n dA,\end{align} is where you tested the laws of physics. The first line was part of your model. The others were just math. And the part where \begin{align}\iint_{S} \left(\vec \nabla \times \vec E\right)\cdot \hat n dA &=\iint_{S} \left(-\frac{\partial \vec B}{\partial t} \right)\cdot \hat n dA,\end{align} was the law of physics.
You use the laws of physics to make models and you compare what happens in the model to what happens in reality. And in this case there isn't a simple thing called an EMF that directly corresponds to some unique curve.
In this model you have electric and magnetic fields everywhere. And in some places there is a wire where the electric field and current density are proportional and everywhere else there is no current density. Then you can make that model conform to Maxwell by making sure the magnetic field and the electric field are connected the way Maxwell demands.
One way to satisfy Maxwell would be to have both the electric and magnetic field to be the electric and magnetic parts of the electromagnetic field given by Jefimenko's equations:
$$\vec E(\vec r,t)=\frac{1}{4\pi\epsilon_0}\int\left[\frac{\rho(\vec r',t_r)}{|\vec r -\vec r'|}+\frac{\partial \rho(\vec r',t_r)}{c\partial t}\right]\frac{\vec r -\vec r'}{|\vec r -\vec r'|^2}\; \mathrm{d}^3\vec{r}' -\frac{1}{4\pi\epsilon_0c^2}\int\frac{1}{|\vec r-\vec r'|}\frac{\partial \vec J(\vec r',t_r)}{\partial t}\mathbb{d}^3\vec r'$$ and $$\vec B(\vec r,t)=\frac{\mu_0}{4\pi}\int\left[\frac{\vec J(\vec r',t_r)}{|\vec r -\vec r'|^3}+\frac{1}{|\vec r -\vec r'|^2}\frac{\partial \vec J(\vec r',t_r)}{c\partial t}\right]\times(\vec r -\vec r')\mathbb{d}^3\vec r'$$ where $t_r$ is actually a function of $\vec r'$, specifically $t_r=t-\frac{|\vec r-\vec r'|}{c}.$
These reduce to Coulomb and Biot-Savart only when those time derivatives are exactly zero, which is statics. So Jefimenko is an example of proper time dependent laws for the electromagnetic field. Note that both the electric and the magnetic part of the electromagnetic field have parts that depend on the time variation of current.
So if you use these equations, the change in current directly causes both electric and magnetic fields. But when the current changes at place-time $(\vec r_1,t_1)$, there is an electric and a magnetic field. But the field exists only at place-times $(\vec r_2,t_2)$ where $t_2=t_1+\frac{|\vec r_2-\vec r_1|}{c}$.
Right from the beginning of studying electromagnetism, I always came before these: straight wire, circular loop, circular ring etcetera. These all referring to 1d current-carrying conductors. See this statement: we need to find the magnetic field at the center due to current in a circular ring- obviously, this is not talking of an annalus - it is talking about a zero-diameter wire, isn't it? That means at all the places, where they are saying current is flowing, $\vec J= \infty$ as the wire diameter is zero. So, does that mean all these books are saying are total wrong?
If you had current flowing through an annulus with inner radius $a$ and outer radius $b$ then when you find the magnetic field in the center, some of the current is a distance $a$ away. Some of it is a distance $b$ away. And some of it is an in between distance away.
But if $(b-a)/b<<1$ then none of the computations are off by very much percentage wise when you are at the center if you just computed as if it was all at $a.$ Divide by $a,$ divide by $b,$ divide by something in between and percentage wise you aren't off by much. The whole idea of a continuous current rather than a velocity times a delta function where each discrete charge is located is a fiction. And trying to have a delta function where each charge is located would be a false accuracy if Quantum Mechanically you don't think discrete charges have a location and a velocity.
So the point is someone could ask you to compute the magnetic field in the center of a circular loop of radius $a$ and if in reality it is super tiny thickness then when you do it properly as an annulus with outer radius $b$ and inner radius $a$ you will get a tiny bit smaller magnetic field. But in the limit as $b$ approaches $a$ you get closer to the result you computed with the 1d circle of current with radius $a.$
But the field near the current itself gets much more accurate when you give it some thickness.
Basically, from far away a thick wire starts to look thin. And if you compute with the thin one you might get real accurate results.
And you might be hung up wrong versus right. Your job is to make predictions. The predictions have to relate to real measurements and the real measurements will always have some uncertainty. Every ruler has a smallest spacing, every clock a shortest tick. Every detector has a rate of misfires that falsely react to nothing, and a rate that it falsely false to react to a true signal. They exist, and so any calculation that starts to get more precise than those errors stops being detectable. And so you never need to be perfect with your theory. You must need to be better than the experimentalist.
And the experimentalist could get arbitrarily good (but still has nonzero error) so you might need to be arbitrarily good too, but you never need to be perfect.
Finally, when percentage wise a much smaller magnetic flux goes through the wires parts than not through the wire, then you aren't very wrong when you just stop at the inner most part of the wire.
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