A plane electromagnetic wave given by E = 85 course bracket 4 into 10 to the power 8 in multiplied by t - 2z travelling in an isotropic magnetic dielectric medium the relative permeability of a medium is given by?

Question

Answer

Understanding the Electromagnetic Wave Equation
The given electromagnetic wave is described by the electric field $ E = 85 \cos(4 \times 10^8 t - 2z) $. This equation indicates a plane wave traveling in the $ z $-direction.

Wave Characteristics
- **Angular Frequency ($ \omega $)**: From the equation, $ \omega = 4 \times 10^8 $ rad/s.
- **Wave Number ($ k $)**: The wave number is $ k = 2 $ rad/m.

Relationship Between Properties
In an isotropic magnetic dielectric medium, the relationship between the speed of light $ c $, the relative permittivity $ \varepsilon_r $, and the relative permeability $ \mu_r $ can be expressed as:
$$
c = \frac{1}{\sqrt{\varepsilon \mu}}
$$
Where $ \varepsilon = \varepsilon_0 \varepsilon_r $ and $ \mu = \mu_0 \mu_r $.

Finding the Speed of Light
The speed of the wave $ v $ can be calculated using:
$$
v = \frac{\omega}{k}
$$
Substituting the values:
$$
v = \frac{4 \times 10^8}{2} = 2 \times 10^8 \text{ m/s}
$$

Relative Permeability Calculation
The speed of light in vacuum is $ c_0 \approx 3 \times 10^8 \text{ m/s} $. The relationship between $ c $ and the medium's properties is:
$$
\frac{1}{\sqrt{\varepsilon_r \mu_r}} = v
$$
Squaring both sides:
$$
\frac{1}{\varepsilon_r \mu_r} = v^2
$$
Thus, rearranging gives:
$$
\mu_r = \frac{1}{\varepsilon_r v^2}
$$
Given that $ v $ is known, if $ \varepsilon_r $ is also known, you can easily compute $ \mu_r $.

Conclusion
To determine the relative permeability $ \mu_r $, the value of relative permittivity $ \varepsilon_r $ is essential. This relationship allows for deeper analysis of the properties of the isotropic magnetic dielectric medium in which the wave propagates.

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