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

 

[HalfLostAndSearching]

I would recommend double-checking your calculations and equations to ensure accuracy. It is possible that the incorrect solution may have a mistake in the calculation or use of equations. Additionally, it may be helpful to physically visualize the rearrangement of the inductor and how it may affect the magnetic field. Another approach could be to compare your solutions to known principles and laws, such as Faraday's Law or the equations for inductance and magnetic field. Ultimately, it is important to carefully analyze and troubleshoot any discrepancies in your solutions to determine the correct answer.