Characteristic Impedance Calculation for Free Space Plane Waves

The question asks for the characteristic impedance of a uniform plane wave specifically in free space. This is a fundamental property related to how electromagnetic waves propagate in a vacuum.

What is Characteristic Impedance?

Characteristic impedance (often denoted by $ \eta $) is the ratio of the electric field strength to the magnetic field strength for a plane wave. It represents the impedance the wave encounters as it travels through a medium. For a uniform plane wave, it's determined by the properties of the medium itself.

Formula for Characteristic Impedance

The general formula for the characteristic impedance of a medium is:

$ \eta = \sqrt{\frac{\mu}{\epsilon}} $

where:

• $ \mu $ is the magnetic permeability of the medium.
• $ \epsilon $ is the electric permittivity of the medium.

Impedance in Free Space

In free space (vacuum), the permeability is $ \mu_0 $ and the permittivity is $ \epsilon_0 $. The standard values are:

• $ \mu_0 = 4\pi \times 10^{-7} \ H/m $ (Henry per meter)
• $ \epsilon_0 \approx \frac{1}{36\pi} \times 10^{-9} \ F/m $ (Farad per meter)

Substituting these values into the formula gives the characteristic impedance of free space, $ \eta_0 $:

$ \eta_0 = \sqrt{\frac{\mu_0}{\epsilon_0}} $

Calculation

Let's perform the calculation:

$ \eta_0 = \sqrt{\frac{4\pi \times 10^{-7} \ H/m}{\frac{1}{36\pi} \times 10^{-9} \ F/m}} $

Simplify the expression:

$ \eta_0 = \sqrt{(4\pi \times 10^{-7}) \times (36\pi \times 10^{9})} $

$ \eta_0 = \sqrt{144\pi^2 \times 10^{2}} $

Take the square root:

$ \eta_0 = \sqrt{(12\pi)^2 \times 10^2} $

$ \eta_0 = (12\pi \times 10) \ \Omega $

$ \eta_0 = 120\pi \ \Omega $

Conclusion

The characteristic impedance of a uniform plane wave in free space is $ 120\pi $ Ohms ($ \Omega $).