a)Propagation constant is given by the formula below
"\\gamma=\\sqrt{({\\dfrac{m\\pi}{a}})^2+({\\dfrac{n\\pi}{b}})^2-\\omega^2\\mu\\epsilon}"
a=0.02m,b=0.01m
For TE10 mode, m=1,n=0
f=50MHz
"\\gamma=\\sqrt{({\\dfrac{1\\pi}{0.02}})^2+({\\dfrac{0\\pi}{0.01}})^2-\\dfrac{(2\\pi\\times{50\\times{10^6}})^2}{(3\\times{10^8})^2}}\\\\ \\gamma=157.1" For TM11 mode
m=1,n=1
"\\gamma=\\sqrt{({\\dfrac{1\\pi}{0.02}})^2+({\\dfrac{1\\pi}{0.01}})^2-\\dfrac{(2\\pi\\times{50\\times{10^6}})^2}{(3\\times{10^8})^2}}\\\\ \\gamma=351.2" For TE20 mode, m=2, n=0
"\\gamma=\\sqrt{({\\dfrac{2\\pi}{0.02}})^2+({\\dfrac{0\\pi}{0.01}})^2-\\dfrac{(2\\pi\\times{50\\times{10^6}})^2}{(3\\times{10^8})^2}}\\\\ \\gamma=314.2" The guide wavelength cannot be calculated without cutoff frequency.
"f_c=\\dfrac{u'}{2}\\sqrt{(\\dfrac{m}{a})^2+(\\dfrac{n}{b})^2}"
For TE10 mode,m=1,n=0
"f_c=\\dfrac{3\\times{10^8}}{2}\\sqrt{(\\dfrac{1}{0.02})^2+(\\dfrac{0}{0.01})^2}=7.5GHz"
for TM11 mode,m=n=1
"f_c=\\dfrac{3\\times{10^8}}{2}\\sqrt{(\\dfrac{1}{0.02})^2+(\\dfrac{1}{0.01})^2}=16.8GHz"for TE20 mode, m=2, n=0
"f_c=\\dfrac{3\\times{10^8}}{2}\\sqrt{(\\dfrac{2}{0.02})^2+(\\dfrac{0}{0.01})^2}=15MHz"
guide wave is given by
"\\lambda_g=\\dfrac{c}{\\sqrt{f^2-f_c^2}}"
Guide wavelength exist for only TE20 mode
"\\lambda_g=\\dfrac{3\\times10^8}{\\sqrt{(50\\times{10^6})^2-(15\\times{10^6})^2}}=6.29m"
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