Why Does Rearranging an Inductor Yield Different Magnetic Fields?

[matej1408]

I found two ways to solve this problem, but I get two different solutions, it's confusing because I can't see the flaw in wrong solution.
1. Homework Statement
Long cylindrical inductor of diameter D₁ and inductance L₁ is connected to battery and creates magnetic field B₁. Inductor is then rearranged to new inductor of diameter D₂ and inductance L₂. Find magnetic field B₂ which is created when "new" inductor is connected to the same battery as before. Assume that length of wire is much larger then length on inductor.

Homework Equations

L=μN²S/l
B=μNi/l
i-current, l-length of coil

The Attempt at a Solution

1st way(correct):[/B]
B₁=μN₁i/l₁ => μN₁/l₁=B₁/i
L₁=μN₁²S₁/l₁
L₁=N₁S₁B₁/i
current is same in both case so:
L₂=N₂S₂B₂/i
dividing and rearranging_
L₁=N₁S₁B₁/i
L₂=N₂S₂B₂/i
B₂=B₁N₁S₁L₂/(N₂S₂L₁)
wire is same length in both case so:
h- thickness of wire
N₁hD₁π=N₂hD₂π
N₁/N₂=D₂/D₁
and S∝D²
B₂=B₁D₁L₂/(D₂L₁)
2nd way:
B_n=μN_ni/l_n => B₂= B₁N₂l₁/(l₂N₁)
L_n=μN_n²S_n/l_n
N_n=√(L_nl_n /(μS_n))
so N₂/N₁=√(L₂l₂S₁/(L₁l₁S₂))
substituting: B₂= B₁√(L₂l₁S₁/(L₁l₂S₂))
wire is same length so: Dπl= constant => l₁/l₂=D₂/D₁
and and S∝D² =>
B₂=B₁√(D₁L₂/(D₂L₁))

 

[TSny]

Hello, and welcome to PF!

1st way(correct):
wire is same length in both case so:
h- thickness of wire
N₁hD₁π=N₂hD₂π

I'm not quite sure why you have a factor of h on each side of the above equation. In any case, it does lead to the following

N₁/N₂=D₂/D₁

which I believe is correct.

Keep in mind that either solenoid can have more than one layer of wraps of wire.

2nd way:
wire is same length so: Dπl= constant => l₁/l₂=D₂/D₁

How is Dπl related to the length of wire?

 

[rude man]

When you connect battery to an inductor the current does not stabilize but ramps up at a constant rate di/dt = V/L. So how a re you comparing currents? the problem also implies that the inductor is changed 'on the fly' while current is flowing, which itself constitutes a V = i dL/dt problem in addition to the usual V = L di/dt one.
Bottom line, the problem is hard to understand, at least for me.

 

[matej1408]

I see where i was wrong, I assumed that there is just one layer but there isn't, thank you TSny.
Coil has some resistance and when is connected to DC there is no voltage change=> current is constant