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Take a infinite continuous solenoid with N turns and length l. We all know that $$B=\frac{\mu_0NI}{l}$$ Several resources said that $B$ $\alpha$ $N$. But i have doubt in it. If number of turns increases in an infinite solenoid doesn't that the length that we consider increase too?

Take it like this. Assume one loop has a length of $l$. The $N$ loops will have length of $Nl$. So in both the cases we get the same results.

So we can conclude that $B$ instead directly proportional to the number of turns per length $n$?

Where i actually went wrong? [Sorry If i asked a terrible question. I'm a newbie to the topic. Thanks for helping!!]

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  • $\begingroup$ What are those resources that say $B\propto N$? $\endgroup$
    – User123
    Commented Sep 17, 2022 at 16:25
  • $\begingroup$ @User123 Well all my text books (Iam in highschool) they have mentioned it. $\endgroup$
    – Sanjay
    Commented Sep 17, 2022 at 16:44

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Your notation is not standard (edit: at least it was not standard before your edit). $N$ is ordinarily considered to be the number of turns, and $n=\frac N l$ is the number of turns per unit length. Using that

$$B=\frac{\mu_0 N I}{l}=\mu_0 n I$$ So that $$B\propto n$$.

So, $B$ is proportial to number of turns per unit length. If $l$ is held constant, $B\propto N$ as well, as your reference might suggest.

The cause of this problem was hence just your misunderstanding of the notation, your reasoning is correct.

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Yes, that's right, magnetic flux density $\mathbf{B}$ is proportional to the number of turns per unit length. If a source says $\mathbf{B}$ is proportional to the number of turns, then they are implicitly holding the length of the solenoid constant.

This is a reasonable assumption; in most applications you would have a fixed core and you would need to decide how many windings to wrap around the core. The windings can be layered on top of each other without invalidating the solution. So even if one winding has length $\ell$ due to the thickness of the wire, you can wind $m$ layers on top of each other to get $m/\ell$ windings per unit length.

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  • $\begingroup$ But if we wind them one over other, doesn't that same logic apply here.... Because the cross-section otherway is also $l$. So... Again they will get cancelled isn't??? Or will we use another method?? $\endgroup$
    – Sanjay
    Commented Sep 17, 2022 at 13:55

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