Question

A 1-MHz uniform plane wave is normally incident onto a fresh water lake (ϵr ′ = 78, ϵr ″ = 0, μr = 1). Determine the fraction of the incident power that is (a) reflected and (b) transmitted. (c) Determine the amplitude of the electric field that is transmitted into the lake.

Answer 1

A. 0.64

B. 0.36

C. 0.203

How to determine this?

Electromagnetic waves travel through different mediums, which can be categorized as dielectric or purely conducting. These waves are composed of plane waves and can propagate through mediums like dielectric substances or free space.

For instance, in air, the index is represented by
`\( n = \sqrt{(\mu_0)/(\epsilon_0)} = 377 \)`.

In a freshwater lake, the index is
`\( n_2 = 377 * \sqrt{(1)/(78)} = 42.68 \)`.

a) To determine the fraction of power reflected, the formula for the fraction of reflected power is given by the square of the reflection coefficient, which is
`\( \left((n_2 - n_1)/(n_2 + n_1)\right)^2 \)`. Substituting the values:
`\( \left((42.68 - 377)/(42.68 + 377)\right)^2 = 0.64 \)`.

b) The fraction of power transmitted can be calculated as
`\( 1 - |\text{Reflection coefficient}|^2 = 1 - 0.64 = 0.36 \)`.

c) Assuming the entire incident amplitude as 1, the amplitude of the electric field transmitted is found using the transmission coefficient formula:

`\( \text{Amplitude} = \text{Incident amplitude} * \text{Transmission coefficient} = 1 * (2n_2)/(n_1 + n_2) = (2 * 42.68)/(42.68 + 377) = 0.203 \, \text{V/m} \)`.