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I'm reading The Art of Electronics(3rd), and I'm just stuck with exercise 1.19. Here is the question:

Explain why L∝n² for an inductor consisting of a coil of n turns of wire, maintaining fixed diameter and length as n is varied.

I'm not expecting a long complete mathematical explaination, because the book only gives me a little information, claiming that that should be enough.

My idea: I∝1/n(because practically a wire does have some resistance), and V∝n. As a result, L=V/I∝n². But I also notice that an ideal wire shouldn't have any resistance. So I'm confused now…

nalzok
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    It has nothing to do with resistance. But a magnetic field. – Eugene Sh. Jul 06 '16 at 16:49
  • If it helps, that 'fixed diameter and length' refers to the coil former the wire is being wound on, **not** to the wire that's being wound. So if n doubles, the length of wire doubles. – Neil_UK Jul 06 '16 at 19:18
  • @EugeneSh. So the current through the inductor doesn't change, and the "back EMF" is increased to n², due to some changes in the magnetic field? – nalzok Jul 06 '16 at 22:59
  • @EugeneSh. **Theoretically**, current I has nothing to do with n, but **practically**, it does(I ∝ 1/n because r ∝ n, where r is neglected in the theory). So the theory tells us L ∝ n², while experiments tell us L ∝ n³? – nalzok Jul 06 '16 at 23:04
  • Hint: more loops - more magnetic field - more energy stored. So I don't know where is your **Theoretically** is coming from. – Eugene Sh. Jul 06 '16 at 23:06
  • @Neil_UK And a double-long wire doubles the magnetic density(just I guess). As a result, the voltage produced on **per length** of the wire is doubled. With the length doubled and voltage on per length doubled, the total back EMF is quadrupled(2²). Am I right? – nalzok Jul 06 '16 at 23:14
  • @EugeneSh. Hmm……Yes, energy is another point of view. I suppose n times more loops products n times stronger magnetic field, and thus n times more energy is stored in it. But how does this lead to n² bigger L? – nalzok Jul 06 '16 at 23:56
  • @NickAlexeev Seems that I 've got it: *magnetic flux* ∝ n, while *magnetic flux linkage* ∝ n². – nalzok Jul 07 '16 at 04:49
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    @sunqingyao Yes, twice as many turns is twice as much flux, twice as many turns is twice as much EMF per flux is 4 times as much EMF per current is 4x as much inductance. – Neil_UK Jul 07 '16 at 07:02

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