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Q.
Electromagnetic waves travel in a medium with a speed of $ 2 \times 10^8 \,m/s $ . If the relative permeability of the medium is $ 1 $ the relative permittvity will be
AMUAMU 2016
Solution:
Given, speed of $EM$ wave $= 2 \times 10^8\, m/s$-
$\mu_r = 1$
$\varepsilon_{r}= ?$
The velocity or speed of $EM$ wave in the medium,
$v =\frac{1}{\sqrt{\mu_{r} \varepsilon_{r}}} \times\frac{1}{\sqrt{\mu_{0} \varepsilon_{0}}}$
We have, $c = \frac{1}{\sqrt{\mu_{0 }\varepsilon_{0}}} $
where $\mu_0$ and $\varepsilon _0$ are permeability and permittivity of
vacuum respectively.
$\therefore v = \frac{1}{\sqrt{\mu_{r} \varepsilon_{r}}} \cdot c$
$ \frac{ 2\times 10^{8}}{3\times 10^{8}} = \frac{1}{\sqrt{\mu_{r} \varepsilon_{r}}} \left[\because C = 3\times10^{8} m s\right] $
$ \frac{2}{3} = \frac{1}{\sqrt{\mu_{r} \varepsilon_{r}}}$
$ \left(\frac{2}{3}\right)^{2} = \frac{1}{\mu_{r} \varepsilon_{r}} = \frac{1}{1 \times \varepsilon_{r}} $
Permittivity, $\varepsilon_{r} = \frac{9}{4} = 2.25$