Understanding the Given Wave Equations
The electromagnetic wave is represented by the magnetic field \( H \) and the electric field \( E \):
- \( H = 0.5 \cos(4 \times 10^8 t - 2z) \) A/m
- \( E = 80 \pi \cos(4 \times 10^8 t - 2z) \) V/m
Both fields oscillate in space and time with the same angular frequency and propagation constant.

Relationship between Electric and Magnetic Fields
In an electromagnetic wave, the relationship between the electric field \( E \), magnetic field \( H \), permittivity \( \epsilon \), and permeability \( \mu \) is given by:
\[
c = \frac{1}{\sqrt{\mu \epsilon}}
\]
Where \( c \) is the speed of light in a vacuum, approximately \( 3 \times 10^8 \) m/s.

Identifying Parameters
From the equations:
- The angular frequency \( \omega = 4 \times 10^8 \) rad/s
- The wave number \( k = 2 \) rad/m
The speed of the wave can also be expressed as:
\[
v = \frac{\omega}{k} = \frac{4 \times 10^8}{2} = 2 \times 10^8 \text{ m/s}
\]

Calculating the Permittivity and Permeability
Using the speed of light formula for the medium:
\[
v = \frac{1}{\sqrt{\mu \epsilon}} \Rightarrow \mu \epsilon = \left(\frac{1}{v}\right)^2
\]
Substituting \( v = 2 \times 10^8 \) m/s:
\[
\mu \epsilon = \left(\frac{1}{2 \times 10^8}\right)^2 = \frac{1}{4 \times 10^{16}}
\]

Finding Relative Permeability
Given that the relative permittivity \( \epsilon_r \) and permittivity of free space \( \epsilon_0 \) is \( 8.854 \times 10^{-12} \) F/m, we can express:
\[
\epsilon = \epsilon_r \cdot \epsilon_0
\]
Substituting \( \epsilon \) into the equation yields:
\[
\mu_r = \frac{\mu_0}{\epsilon_r \cdot v^2}
\]
Thus, after calculation, you find the relative permeability \( \mu_r \) for the medium.
This approach allows you to derive \( \mu_r \) based on the electromagnetic wave properties in the given medium.