Finding the resonant frequency of a rectangular resonator filled with a magnetic material

Question

The prompt is to find the resonant frequency $f_r$ of a rectangular resonator which is filled with a magnetic material rather than standard air or vacuum. I'm confused as how the resonance frequency changes when its filled with a magnetic material.

Noting that the walls are PEC.
TM mode is $TM_{110}$
$$epsilon_r = 1  mu_r = 1  sigma = 0 s/m $$
$$ a = b = Xmm  c = Ymm  (X &Y) > 0$$

My approach:
$$ 
f_{110} = frac{c}{2pi} sqrt{ bigl(frac{ pi}{a}bigr)^2 + bigl(frac{pi}{b}bigr)^2} $$
$$ f_{110} = frac{1}{v.2 pi}sqrt{ bigl(frac{ pi}{a}bigr)^2 + bigl(frac{pi}{b}bigr)^2} $$

$$ f_{110} = frac{1}{sqrt{epsilonepsilon_r. mu mu_r }}sqrt{ bigl(frac{ pi}{a}bigr)^2 + bigl(frac{pi}{b}bigr)^2} $$

Assuming the speed of light is lesser in a magentic medium.

source : https://www.ee.iitb.ac.in/uma/~rkashyap/ee614rep.pdf - eqn(20)

freecharly · Answer

Your answer seems to be correct. The resonant frequencies depend on the speed $v$ of the electromagnetic wave in the medium inside the resonator $$v=frac {c}{sqrt{epsilon_r mu_r}}$$ so that you can get the resonant frequencies for a $mu_r neq 1$ and $epsilon_r=1$ by substituting $c$ by $v$ in the relevant formulas as long as you have a linear paramagnetic ($mu gt 1$) or diamagnetic ($mu lt 1$) medium. Ferromagnetic materials can, in general, not be considered to be linear.

Answered by freecharly on April 28, 2021

Finding the resonant frequency of a rectangular resonator filled with a magnetic material

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