Two coils of wire with the same size and shape

[cdummie]

Homework Statement

Two coils of wire that have same shape and dimensions are thickly rolled up so that the coincide, they only differ in number of coils (windings) N1 for the firs one and N2 for the second one (N1<N2). When there's constant current in the first one and there's no current in the second one then fluxes through those two coils are the same. Determine the expression for coupling coefficient of these two coils.

Homework Equations

$k=\frac{ |L_{12}| }{\sqrt{L_1 L_2}}$

The Attempt at a Solution

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In case when there's no current in the second coil and there is constant current in the second coil then there's no magnetic induction vector coming from the second coil so only flux second coil has is flux coming from the first coil $Ф_{12}=N_2L_{12}I_{1}$, while flux in the first coil is the flux coming from it's own magnetic induction vector (it's own current) and it's value is $Ф_1=N_1L_1I_1$.

Since these two are equal it means that $N_2L_{12}I_{1}=N_1L_1I_1 \Rightarrow N_2L_{12}=N_1L_1 \Rightarrow L_{12}=\frac{N_1L_1}{N_2}$

But i don't know how to find $k=\frac{|L_{12}|}{\sqrt{L_1 L_2}}$ because this tells me nothing about $L_2$. Anyone knows what can i do here?

 

[NascentOxygen]

Does this require you to use the formula for inductance of a solenoid having area, length, and number of turns?

 

[rude man]

What is the definition of k really? It's not basically M/√(L1⋅L2) although here the relation happens to be correct.
Define k in terms of the flux generated in coil 1 for a given current I, and the flux coupling into coil 2. The problem tells you the answer! Then compute k.

 

[cdummie]

Does this require you to use the formula for inductance of a solenoid having area, length, and number of turns?

No, there's no special requirements, it only says that i should determine k.

 

[cdummie]

Does this require you to use the formula for inductance of a solenoid having area, length, and number of turns?

There's no any other requirements, it just says to find k.

 

[cdummie]

What is the definition of k really? It's not basically M/√(L1⋅L2) although here the relation happens to be correct.
Define k in terms of the flux generated in coil 1 for a given current I, and the flux coupling into coil 2. The problem tells you the answer! Then compute k.

flux in the first coil is $Ф_1=N_1L_1I_1$ and flux in second coil is $Ф_2=N_2L_{12}I_1$ but how can i represent k in terms of these values?

 

[rude man]

flux in the first coil is $Ф_1=N_1L_1I_1$ and flux in second coil is $Ф_2=N_2L_{12}I_1$ but how can i represent k in terms of these values?

You don't.
There is a much more basic definition of k. Hint: all you need is Φ₁₁ and Φ₂₁. Forget the L's, I's. and N's.

Φ₁₁ is the flux generated by coil 1 coupling into coil 1, Φ₂₁ is the flux generated by coil 1 coupling into coil 2. What does the problem say about those two quantities?

 

[rude man]

After re-reading the problem I see it's an incomplete statement. You could conceivably have coil1 coupling fully into coil 2 without coil 2 coupling fully into coil 1. However, I'm guessing they meant that both coils couple into each other 100%. Otherwise there are really two coupling coefficients and the problem would be unsolvable.

 

[cdummie]

You don't.
There is a much more basic definition of k. Hint: all you need is Φ₁₁ and Φ₂₁. Forget the L's, I's. and N's.

Φ₁₁ is the flux generated by coil 1 coupling into coil 1, Φ₂₁ is the flux generated by coil 1 coupling into coil 2. What does the problem say about those two quantities?

I've tried to find that more basic definition of k in my books and notes, but i couldn't, but i found something that can be related to this problem, i found out that if whole flux through first coil goes through second coil (through it's cross section to be precise) that then we have perfect coupling. If this is correct, it means that in this case, k=1,but , is there any way i could algebraically prove this?

 

[rude man]

I've tried to find that more basic definition of k in my books and notes, but i couldn't, but i found something that can be related to this problem, i found out that if whole flux through first coil goes through second coil (through it's cross section to be precise) that then we have perfect coupling. If this is correct, it means that in this case, k=1,but , is there any way i could algebraically prove this?

This is the correct answer. You can't prove what is defined! k = fraction of flux generated by coil 1 coupled into coil 2.

You should be aware that you can have 100% of the flux from coil 1 coupled into coil 2, but less than 100% from coil 2 to coil 1. I recommend my "Insight" paper discussing this more fully: https://www.physicsforums.com/insights/misconceiving-mutual-inductance-coefficients/

 

[cdummie]

This is the correct answer. You can't prove what is defined! k = fraction of flux generated by coil 1 coupled into coil 2.

You should be aware that you can have 100% of the flux from coil 1 coupled into coil 2, but less than 100% from coil 2 to coil 1. I recommend my "Insight" paper discussing this more fully: https://www.physicsforums.com/insights/misconceiving-mutual-inductance-coefficients/

Thank you for your help. I'll check out that "Insight" paper of yours.