# GATE problems in EMT

pb. An Electro Magnetic Wave Propagates through a loss less insulator with a velocity $1.5&space;X&space;10^{10}$ Cm/sec . Calculate the electric magnetic properties of the insulator if its intrinsic impedance is $90\pi$ Ohms.

Ans:- Given loss less insulator  $\Rightarrow&space;\alpha&space;=0$

$v_{p}&space;=1.5&space;X&space;10^{10}$ Cm/Sec

$v_{p}&space;=&space;\frac{1}{\sqrt{\mu&space;_{o}\epsilon&space;_{o}\mu&space;_{r}\epsilon&space;_{r}}}$

$1.5&space;X&space;10^{10}=&space;\frac{1}{\sqrt{\mu&space;_{o}\epsilon&space;_{o}\mu&space;_{r}\epsilon&space;_{r}}}$

$1.5&space;X&space;10^{10}=&space;\frac{3&space;X&space;10^{10}}{\sqrt{\mu&space;_{r}\epsilon&space;_{r}}}$

${\sqrt{\mu&space;_{r}\epsilon&space;_{r}}}&space;=&space;2&space;------------EQNI$

from $\eta&space;=\sqrt{\frac{\mu&space;_{o}\mu&space;_{r&space;}}{\epsilon&space;_{o}\epsilon&space;_{r&space;}}}$

$90\pi&space;=&space;\sqrt{\frac{\mu&space;_{o}\mu&space;_{r&space;}}{\epsilon&space;_{o}\epsilon&space;_{r&space;}}}$

$90\pi&space;=&space;\sqrt{\frac{\mu&space;_{r&space;}}{\epsilon&space;_{r&space;}}}&space;.&space;120&space;\pi$

$\sqrt{\frac{\mu&space;_{r&space;}}{\epsilon&space;_{r&space;}}}&space;=\frac{3}{4}--------EQNII$

from Equations I and II $\mu&space;_{r}&space;=&space;\sqrt{\frac{\mu&space;_{r&space;}}{\epsilon&space;_{r&space;}}}&space;\sqrt{\mu&space;_{r&space;}\epsilon&space;_{r}}$

$\mu&space;_{r}&space;=&space;2&space;X&space;\frac{3}{4}$

$\mu&space;_{r}&space;=&space;1.5$

$\sqrt{\mu&space;_{r&space;}\epsilon&space;_{r&space;}}&space;=&space;2$

$\sqrt{1.5&space;\epsilon&space;_{r&space;}}&space;=&space;2$

$\epsilon&space;_{r}&space;=&space;2.66$

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