The standing-wave ratio (SWR) is a measure of the impedance mismatch between two media and is given by the ratio of the maximum amplitude of the standing wave to the minimum amplitude. To calculate the SWR in this case, we need to consider the reflection coefficient at the interface between air and the nonmagnetic lossless medium.
The reflection coefficient (Γ) can be calculated using the formula:
Γ = (Z2 - Z1) / (Z2 + Z1)
Where Z1 and Z2 are the characteristic impedances of the two media. In this case, air is the first medium and the nonmagnetic lossless medium is the second medium.
The characteristic impedance of air (Z1) is approximately equal to the impedance of free space, which is around 377 ohms.
The characteristic impedance of the nonmagnetic lossless medium (Z2) is given by:
Z2 = sqrt(μ0 / εr) * Z0
Where μ0 is the permeability of free space (4π × 10^-7 T·m/A), εr is the relative dielectric constant of the medium (given as 25 in this case), and Z0 is the impedance of free space.
Plugging in the values, we have:
Z2 = sqrt((4π × 10^-7 T·m/A) / 25) * 377 ohms
Calculating Z2, we get:
Z2 ≈ 16.927 ohms
Now, we can calculate the reflection coefficient Γ:
Γ = (16.927 ohms - 377 ohms) / (16.927 ohms + 377 ohms)
Simplifying the expression, we have:
Γ ≈ -0.978
The standing-wave ratio (SWR) is related to the reflection coefficient by the formula:
SWR = (1 + |Γ|) / (1 - |Γ|)
Plugging in the value of Γ, we get:
SWR = (1 + |-0.978|) / (1 - |-0.978|)
Simplifying the expression, we have:
SWR ≈ 47.062
Therefore, the standing-wave ratio in the air is approximately 47.062.