Magnetic field produced by a solenoid

[Nabeshin]

Ok, so anyone who has studied magnetism knows that the magnetic field due to a solenoid is given by the equation B=\mu_oNI Where N is the number of turns in a given length. Well, in thinking about these, I tend not to make measurements and would rather like to predict the magnetic field of the solenoid, so I tried to "simplify" the formula. Here goes:

B=\mu_oNI
N is turns, which I call n per length, which I will call L_s
B=\frac{\mu_onI}{L_s}
From ohms law, I = V/R
B=\frac{\mu_onV}{L_sR}
And R is equal to \frac{\rho L_w}{A} Where L_w is the length of wire and A is its cross-sectional area.
B=\frac{\mu_onVA}{\rho L_s L_w}

Up till here I'm confident, but after this I'm not so sure. Now, I tried to model the helical nature of the wire wrap by creating a vector-valued function:
r(t)=sin(t)i-cos(t)j+\frac{r_w}{\pi}t k\left
Now in doing this I assume that the coils are wrapped as tight as possible. By this I mean that the horizontal spacing between two loops is equal to 2r, or the diameter of the wire. So this function should move a distance of 2r up for every turn (2*pi radians).

Now, arc length is given by the formula: s(t)=\int||r'(t)||dt So...

s(t)=\int\sqrt{sin^{2}(t)+cos^{2}(t)+(\frac{r_w}{\pi})^{2}}dt
Now, in one turn of the wire we rotate 2pi radians, so let's evaluate from 0 to 2pi..
s(t)=\int^{2\pi}_{0}\sqrt{sin^{2}(t)+cos^{2}(t)+(\frac{r_w}{\pi})^{2}}dt=2\sqrt{\pi^{2}+r^{2}}
So this is the length of wire per one turn, so we can say:

\frac{Length\<img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f631.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":o" title="Eek! :o" data-smilie="9"data-shortname=":o" />f\:wire}{Turn\<img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f631.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":o" title="Eek! :o" data-smilie="9"data-shortname=":o" />f\:wire}=2\sqrt{\pi^{2}+r^{2}}

And turns of wire is n, so we finally arrive at:

L_w = 2n\sqrt{\pi^{2}+r^{2}}

Back to the main mission! Substituting in for L_w produces:

B=\frac{\mu_o A V}{2\rho L_s\sqrt{\pi^{2}+r^{2}}}

So I gather that the magnetic field of a solenoid depends on three things: How long the solenoid is, the specific wire you're using (both size and material), and the voltage applied. Well, does this make sense to anyone?

 

[LURCH]

So I gather that the magnetic field of a solenoid depends on three things: How long the solenoid is, the specific wire you're using (both size and material), and the voltage applied. Well, does this make sense to anyone?

Accept that you left out the fourth thing. Strengthen the magnetic field is a function of "field density." And the density of the field is determined by the number of turns in the coil per inch of links along the coil. So, the field strengths does not only depend on the length of the coil, but also the density of the windings. Unless, of course, you're talking about a standardized winding density, in which case length of the coil is all that matters.

 

[Nabeshin]

Accept that you left out the fourth thing. Strengthen the magnetic field is a function of "field density." And the density of the field is determined by the number of turns in the coil per inch of links along the coil. So, the field strengths does not only depend on the length of the coil, but also the density of the windings. Unless, of course, you're talking about a standardized winding density, in which case length of the coil is all that matters.

Well, I assumed the coils were right next to each other which is as tight as they can get, which I think is what you mean by standardized winding density.