Inductance Puzzle involving coplanar conducting rings

In summary, the "Inductance Puzzle involving coplanar conducting rings" explores the behavior of two or more conductive rings placed in the same plane and their mutual inductance effects. It examines how changes in current through one ring can induce an electromotive force (EMF) in the others, leading to various intriguing phenomena. The puzzle highlights the complexities of calculating inductance in such configurations, considering factors like geometry, distance, and the material properties of the rings. This scenario serves to illustrate fundamental principles of electromagnetism and inductance in electrical circuits.
  • #1
Hak
709
56
Homework Statement
We have a small ring made of thin wire having radius ##R## and its inductance is ##L##. Find the inductance of a ring having ##n##-times the dimensions as this ring. If in the plane of the ring, we place another superconducting ring of half the geometric dimensions so that the planes of the rings and their centers coincide, then the inductance of the ring with radius ##R## comes out to be ##L_1##. What will the inductance ##L_2## of the ring with radius ##R## be when it is placed inside a superconducting ring with twice the geometric dimensions? The planes and centers of the rings also coincide in this case.
Relevant Equations
/
I solved the first point as ##L_n = \frac{\mu_0 n R}{2}##. How to proceed for the other point?
 
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  • #2
Hak said:
Homework Statement: We have a small ring made of thin wire having radius ##R## and its inductance is ##L##. Find the inductance of a ring having ##n##-times the dimensions as this ring. If in the plane of the ring, we place another superconducting ring of half the geometric dimensions so that the planes of the rings and their centers coincide, then the inductance of the ring with radius ##R## comes out to be ##L_1##. What will the inductance ##L_2## of the ring with radius ##R## be when it is placed inside a superconducting ring with twice the geometric dimensions? The planes and centers of the rings also coincide in this case.
Relevant Equations: /

I solved the first point as ##L_n = \frac{\mu_0 n R}{2}##. How to proceed for the other point?
Welcome to PF.

Please show us your work on the first question, so we can understand your approach. Thanks.
 
  • #3
Let us recall the formula for the inductance of a ring of radius ##R## and wire cross-sectional area ##A##, as given by ##L = \frac{\mu_0 N^2A}{l}##, where ##\mu_0## is the permeability of free space, ##N## is the number of turns of the wire, and ##l## is the length of the wire. For a single-turn ring, we have ##N = 1## and ##l = 2 \pi R##, so the formula simplifies to ##L = \frac{\mu_0 A}{2 \pi R}##. Since ##A = A_{circle} = \pi R^2##, we have: ##L = \frac{\mu_0 R}{2}##.

Now, if we have another ring that has ##n##-times the dimensions of the original ring, that means its radius is ##R_n = nR## and its cross-sectional area is ##A_n = \pi n^2 R^2 = n^2 A##. Plugging these values into the formula, we get: $$L_n = \frac{\mu_0 A_n}{2 \pi n R} = \frac{\mu_0 \pi n^2 R^2}{2 \pi n R} \Rightarrow L_n = \frac{\mu_0 n R}{2} = nL$$
 
  • #4
Is it correct?
 
  • #5
Hak said:
Is it correct?
Could someone please tell me whether this result is correct? In that case, I would post my approach to calculate ##L_1##, although I firmly believe it is wrong, as it does not lead to ##L_2##.
 
  • #6
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FAQ: Inductance Puzzle involving coplanar conducting rings

What is the basic principle behind inductance in coplanar conducting rings?

The basic principle behind inductance in coplanar conducting rings is electromagnetic induction. When the magnetic flux through a conducting ring changes, it induces an electromotive force (EMF) in the ring according to Faraday's Law of Induction. This induced EMF generates a current if the ring is part of a closed circuit. The mutual inductance between two coplanar rings depends on their relative positions and orientations.

How does the distance between the rings affect their mutual inductance?

The mutual inductance between two coplanar conducting rings decreases as the distance between them increases. This is because the magnetic field produced by the current in one ring weakens with distance, resulting in a smaller magnetic flux through the second ring. Consequently, the induced EMF and the mutual inductance are reduced.

Can the orientation of the rings influence their inductance?

Yes, the orientation of the rings significantly influences their mutual inductance. If the rings are aligned such that their planes are parallel and their centers are aligned, the mutual inductance is maximized. If the rings are tilted or misaligned, the mutual inductance decreases because the effective magnetic flux linkage between the rings is reduced.

What role does the frequency of the current play in the inductance of coplanar rings?

The frequency of the current affects the inductive reactance of the rings, which is the opposition to the change in current due to inductance. Higher frequencies increase the inductive reactance, making it more difficult for the current to change rapidly. However, the mutual inductance itself, which is a geometric property, is not directly affected by the frequency of the current.

How can we calculate the mutual inductance between two coplanar conducting rings?

The mutual inductance between two coplanar conducting rings can be calculated using the Neumann formula, which involves integrating the magnetic field produced by one ring over the area of the second ring. This calculation typically requires knowledge of the rings' radii, the distance between them, and their relative orientation. For simple configurations, approximate analytical expressions or numerical methods can be used to determine the mutual inductance.

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