How Does Coil Configuration Affect Solenoid Inductance?

[Rayanna]

Two coils are made of copper wires of same length .In the first coil number of turns is 3n and radius is r . In the second coil number of turns is n and radius is 3r the ratio of self inductances of the coil is:

I know that self inductance of a solenoid is μN²A/l ;
where A = area of cross section of frame of solenoid ;so by puting values I get 1:1 which is incorrect ? Please explain me where I am wrong as the correct answer is 3:1.

 

[PeroK]

Two coils are made of copper wires of same length .In the first coil number of turns is 3n and radius is r . In the second coil number of turns is n and radius is 3r the ratio of self inductances of the coil is:

I know that self inductance of a solenoid is μN²A/l ;
where A = area of cross section of frame of solenoid ;so by puting values I get 1:1 which is incorrect ? Please explain me where I am wrong.

It's a sort of trick question. Hint: what is "length" in this context?

 

[Rayanna]

It's a sort of trick question. Hint: what is "length" in thi

It's a sort of trick question. Hint: what is "length" in this context?

I am not sure but may be copper wires are of same length.

But then I can't find the length of solenoid

 

[PeroK]

$l$ is the length of the solenoid. Is that the same in both cases? The total length of the wire is the same.

Also, what is $N$? Is that the same in both cases?

 

[Rayanna]

$l$ is the length of the solenoid. Is that the same in both cases? The total length of the wire is the same.

Also, what is $N$? Is that the same in both cases?

I am also not sure that length of solenoid is same, as it is not cleary stated in question that whether copper wires used to make coil are of same length or coils are of same length but N in first case is 3n and in second case is n.

 

[PeroK]

I am also not sure that length of solenoid is same, as it is not cleary stated in question that whether copper wires used to make coil are of same length or coils are of same length but N in first case is 3n and in second case is n.

You are told that the wire is the same length in both cases, and I think you are supposed to assume that the wire is coiled with the same number of coils per unit length in both cases. This would give you:

$r_1 = r, \ N_1 = 3n, \ l_1 = l$

$r_2 = 3r, \ N_2 = n, \ l_2 = l/3$

The solenoid must be shorter in th second case, where the length is each coil is longer.

 

[Rayanna]

You are told that the wire is the same length in both cases, and I think you are supposed to assume that the wire is coiled with the same number of coils per unit length in both cases. This would give you:

$r_1 = r, \ N_1 = 3n, \ l_1 = l$

$r_2 = 3r, \ N_2 = n, \ l_2 = l/3$

The solenoid must be shorter in th second case, where the length is each coil is longer.

Thanks for the solution.
But still I did not understand the point that why we should assume that number of turns per united length is same?

 

[PeroK]

Thanks for the solution.
But still I did not understand the point that why we should assume that number of turns per united length is same?

If you make the solenoid the same length in both cases, then your 1-1 ratio is correct. It might seem quite natural to do this in order to comapre the inductance in the two cases.

But, another natural approach is to coil the wire equally in both cases (probably tightly). This leads to different lengths of solenoid in the two cases.

On reflection, I would say the second assumption is better, because you can definitely do that. In the first case, you may have to stretch the wire and the loops are less and less like circles if the radius is large.

A diagram might help.

 

[Rayanna]

If you make the solenoid the same length in both cases, then your 1-1 ratio is correct. It might seem quite natural to do this in order to comapre the inductance in the two cases.

But, another natural approach is to coil the wire equally in both cases (probably tightly). This leads to different lengths of solenoid in the two cases.

On reflection, I would say the second assumption is better, because you can definitely do that. In the first case, you may have to stretch the wire and the loops are less and less like circles if the radius is large.

A diagram might help.

Thank you.

I get it so we have to assume both copper wires are of same length and we have coiled them in such manner that there number of turns per unit length are same.