Derivation of magnetic field of a Solenoid: Biot savart law

[Conductivity]

Hello,

I have seen that biot savart's law works for infinitely narrow wires:
"The formulations given above work well when the current can be approximated as running through an infinitely-narrow wire."

When I wanted to derive the magnetic field of a solenoid, I had to do this substitution:
$n_o = N/L$

$k = n_o dx$
Where k is the number of turns per dx.. But shouldn't K be an integer? so I can substitute it in the formula for circular coils. That means I have infinite number of turns and turn density of something like $\frac{a}{dx}$ where a is an integer.

Is there is something wrong or that this is the idealization that we do to the solenoid? Wouldn't it be way off the correct value?

If you want the proof, http://nptel.ac.in/courses/122101002/downloads/lec-15.pdf
Page 8, Example 9.

Thank you in advance.

 

[BvU]

But shouldn't K be an integer?

No reason. If you have an 11 cm coil with 10 turns, you have 90.91 turns/m.

Note: never, never ever write something like "$\frac{a}{dx}$ with $a$ finite".
$dx$ is universally seen as an infinitesimal (something that goes to zero). So $\frac{a}{dx}$ does not exist.

 

[Conductivity]

No reason. If you have an 11 cm coil with 10 turns, you have 90.91 turns/m.

Don't we idealize a solenoid as a number of circular coils?

https://i.imgur.com/RCO3qcQ.png

If we take a dx piece of this solenoid and treat at as k number of coils,

The equation for a single coil is:
$B = \frac{ u_o i R^2}{2 ( R^2 + x^2)^{\frac{3}{2}}}$
If dx has 2 turns then we multiply by 2, If it has k turns then we multiply by k. But a dx piece can't have a 2.5 or a fraction of a coil (It doesn't make sense), Can it?
That is why I said $k = n_o dx$ has to be an integer.

 

[BvU]

Don't we idealize a solenoid as a number of circular coils?

We calculate the B field as if it were caused by a cylindrical sheet of current. That is a very good approximation (example 8 already indicates that).

If dx has 2 turns then we multiply by 2, If it has k turns then we multiply by k. But a dx piece can't have a 2.5 or a fraction of a coil

Again: do NOT use infinitesimals this way -- it will get you into trouble. A infinitesimal $dz$ piece has zero turns and nevertheless contributes to B with an infinitesimal contribution $dB$ proportional to ${N\over L}$. It is only when you integrate $dB$ over a finite range in $z$ that you get a finite result. The derivation (it's not a proof) in the pdf is just fine.