Inductive reactance – Circular loop with N lambda standing wave

[Sandgroper]

Has anyone come across, or may be able to point me in the direction of a method for evaluating the inductive reactance of a circular loop when the wavelength of the applied signal is significantly less than the conductor length of the loop - and more particularly when;

A. The driven wavelength is an even whole multiple of the loop diameter, and
B. The loop forms part of a ‘long’ leg in a parallel resonant circuit carrying a standing wave?

I ask this question to the general physics community as it has both a bearing on the work of Faraday, Maxwell and Einstein in terms of fundamental electromagnetic field phenomena, as well as a practical application in the production and analysis of particular magnetic fields.

Inductive reactance is well understood to be a vector function of dPhi/dt and can be readily calculated and observed for parallel conductors, single loop and multi loop coils through to odd shaped coils thanks to Msrs Wheeler et. al. etc. Common to all approaches I have come across is that the applied wavelength is significantly less than the conductor length of the coil (Actually the wavelength << conductor length is implicit in the formulas). The literature suggests that as the geometry of a coil increases the inductance, and hence the inductive reactance approach infinity - this to me seems to be predicated on the assumption that the wavelength of the applied signal is significantly less than the guided path.

I can not find a generic or fundamental method of calculating inductance/reactance for cases where the wavelength is significantly shorter than the loop/coil/conductor/guide length.

Any help will be greatly appreciated.

 

[eljose79]

Thank you for bringing up this interesting question about the inductive reactance of a circular loop in different scenarios. As a scientist in the field of electromagnetism, I may be able to provide some insight and direction for your inquiry.

Firstly, let's address the issue of the applied wavelength being significantly shorter than the loop length. In this case, the loop can be considered as a short dipole antenna, and the inductive reactance can be calculated using the formula Xl = 2*pi*f*L, where f is the frequency of the applied signal and L is the inductance of the loop. This formula is valid for loops of any shape, including circular loops.

Moving on to the specific scenarios you mentioned - when the driven wavelength is an even whole multiple of the loop diameter and when the loop forms part of a parallel resonant circuit carrying a standing wave - the calculation of inductive reactance becomes more complex. In these cases, the loop can be considered as a transmission line, and the inductive reactance can be calculated using transmission line theory and the Smith chart.

The Smith chart is a graphical tool used to analyze transmission line problems, and it can be used to calculate the inductive reactance of a circular loop in these scenarios. You can find more information on the Smith chart and its applications in electromagnetism textbooks or online resources.

Additionally, there are also numerical methods such as finite element analysis that can be used to calculate the inductive reactance of a circular loop in these scenarios. However, these methods may require specialized software and expertise.

I hope this information helps guide you in the right direction for your research. If you have any further questions or need clarification, please do not hesitate to reach out. Best of luck in your studies.