Why does the phi component of the current is zero?

Aug 2019
88
23
India
These are images from the book Introduction to Electrodynamics by David J. Griffiths .

mymath.png.


mymath2.png.

My problem is that I'm unable to understand how the current has zero $\phi$ component (I have underlined it in the first image)? I do understand cylindrical coordinates, I know cylindrical coordinates involve three components $(r, \phi, z)$ and $\hat{r}$ points radially outwards, $\hat{\phi}$ points perpendicular to $\hat{r}$ and even to $z$ axis.

I fully understand this image (credit: Frobenius)
1584382152017.png.

But in spite of all these I can't seem to understand how the current have no $\phi$ component, Well, in that toroid we have current going on in circles and really I don't anything more than this.

I tried making some diagrams for understanding it, like these
1584382273066.png.

1584382288361.png.

I asked some people and they have this to say
There will always be a $\phi$ component of II from the toroid, but if $n=N/L$ is made very large, then the toroid current $I$ can be minimized. This step is really necessary, because it really simplifies things.
Theoretically, we can also make the wire as fine as possible. The result is a current per unit length exactly how it is modeled, without any $\phi$ component, with $n=N/L \to ~+\infty$ , and $I \to 0$, and finite current per unit length $K=nI$
Please help me, devise your own method to explain me. I have tried so so hard to understand them but to be honest it seems to me that the things that I have quoted above have no relation to what I asked. Please help me.
 

topsquark

Math Team
May 2013
2,532
1,051
The Astral plane
We are looking at the situation as if we have a huge number of individual loops of wire that are independent of each other. Take the plane of one of these coils as being in the plane formed by, say the xz plane. The loop has no width to it so it has no span in the \(\displaystyle \phi\) direction. If we had only a few loops with a physical wire (that is to say it has a width) we couldn't say this.

-Dan
 
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Aug 2019
88
23
India
We are looking at the situation as if we have a huge number of individual loops of wire that are independent of each other. Take the plane of one of these coils as being in the plane formed by, say the xz plane. The loop has no width to it so it has no span in the \(\displaystyle \phi\) direction. If we had only a few loops with a physical wire (that is to say it has a width) we couldn't say this.

-Dan
Thank you so much, you solved my doubt very efficiently.