An electromagnetic wave is propagating through a medium with...

First, find the wave impedance ($Z$) of the medium, which is the ratio of the electric field to the magnetic field ($E/H$):
$Z = \sqrt{\dfrac{\mu}{\varepsilon}} = \sqrt{\dfrac{\mu_0 \mu_r}{\varepsilon_0 \varepsilon_r}}$

$Z = \sqrt{\dfrac{\mu_0}{\varepsilon_0}} \times \sqrt{\dfrac{\mu_r}{\varepsilon_r}}$

Substitute the given values:
$Z = 120 \pi \times \sqrt{\dfrac{1}{4}} = 120 \pi \times \dfrac{1}{2} = 60 \pi \, \Omega$

The average intensity ($I$) of an electromagnetic wave is related to the rms electric field ($E_{\text{rms}}$) and the wave impedance ($Z$) by the formula:
$I = \dfrac{E_{\text{rms}}^2}{Z}$

Substitute the known values of intensity and impedance:
$\dfrac{540}{\pi} = \dfrac{E_{\text{rms}}^2}{60 \pi}$

Multiply both sides by $60 \pi$:
$E_{\text{rms}}^2 = \dfrac{540}{\pi} \times 60 \pi$

$E_{\text{rms}}^2 = 32400$

Taking the square root gives:
$E_{\text{rms}} = 180 \text{ V/m}$

Answer: $180 \text{ V/m}$