We know that the electromagnetic wave propagates perpendicular to both electric field and magnetic field which can describe as
$\overrightarrow{k} \times \overrightarrow{E}= \omega \overrightarrow{B} \qquad(1)$
If $\hat{n}$ is a unit vector in the direction of the propagation then
$\overrightarrow{k}=k \hat{n}$
Substitute these values in equation$(1)$ then we get
$k(\hat{n} \times \overrightarrow{E})= \omega \overrightarrow{B}$
$\overrightarrow{B}= \frac{k}{\omega}(\hat{n} \times \overrightarrow{E}) \qquad(2)$
But the value of $k$ and $\omega$ is
$k=\frac{2\pi}{\lambda}$
$\omega=2 \pi \nu$
Then value of $\frac{k}{\omega}=\frac{1}{c}$
Now substitute the value of $\frac{k}{\omega}$ in equation$(2)$ then we get
$\overrightarrow{B}= \frac{1}{c}(\hat{n} \times \overrightarrow{E})$
The magnitude form of the above equation can be written as
$B=\frac{E}{c}$
$\frac{E}{\mu_{0}H}=c \qquad (\because B=\mu_{0} H)$
$\frac{E}{H}=\mu_{0}c$
$\frac{E}{H}=\frac{\mu_{0}}{\sqrt{\mu_{0} \epsilon_{0}}} \qquad(\because c=\frac{1}{\sqrt{\mu_{0} \epsilon_{0}}})$
$\frac{E}{H}= \sqrt{\frac{\mu_{0}}{\epsilon_{0}}}$
The term $\frac{E}{H}$ has dimensions of the impedance and is known as characteristic impedance or intrinsic impedance of free space. It is represented by $(Z_{0})$.
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$Z_{0}=\frac{E}{H}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}$
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Now substitute the value of $\mu_{0}$ and $\epsilon_{0}$ i.e.
$\mu_{0}=4\pi \times 10^{-7}$
$\epsilon_{0}=8.854 \times 10^{-12}$
$Z_{0}=\sqrt{\frac{4\pi \times 10^{-7}}{8.854 \times 10^{-12}}}$
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$Z_{0}=120\pi \: \Omega $
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