A uniform plane wave at 500 MHz operating frequency is incident from medium 1 (Taconic TLX0) with er=2.45 to medium 2 (FR4) with e2=4.3. The direction of propagation is along the z-axis. The electric field is TM polarised an amplitude of 250 V/m. The angle of incidence is 45°. Calculate the following: (i) The angle of transmission, (ii) the reflection and transmission coefficients, (iii) the time domain expressions for the transmitted magnetic fields. (20 marks)

(i) The angle of transmission:

The angle of transmission (θt) can be calculated using Snell's Law, which states that the ratio of the sine of the angle of incidence (θi) to the sine of the angle of transmission is equal to the ratio of the speed of light in the first medium to the speed of light in the second medium. This can be expressed as:

sin(θi) / sin(θt) = sqrt(er1) / sqrt(er2)

Given that the angle of incidence is 45° and the relative permittivities of the two media are er1 = 2.45 and er2 = 4.3, we can substitute these values into the equation to find:

sin(45°) / sin(θt) = sqrt(2.45) / sqrt(4.3)

Solving for θt gives:

θt = arcsin [ sin(45°) * sqrt(4.3) / sqrt(2.45) ]

(ii) The reflection and transmission coefficients:

The reflection coefficient (Γ) and transmission coefficient (T) can be calculated using the Fresnel equations. For TM polarization, these are given by:

Γ = (Z2cos(θi) - Z1cos(θt)) / (Z2cos(θi) + Z1cos(θt))

T = 2Z2cos(θi) / (Z2cos(θi) + Z1cos(θt))

where Z1 and Z2 are the characteristic impedances of the two media. These can be calculated using the formula:

Z = sqrt(μ/ε)

where μ is the permeability and ε is the permittivity of the medium. For most materials, μ is approximately equal to the permeability of free space (μ0), so the characteristic impedance is primarily determined by the relative permittivity (er).

(iii) The time domain expressions for the transmitted magnetic fields:

The time domain expression for the magnetic field (H) of a plane wave is given by:

H(t) = H0cos(ωt - βz + φ)

where H0 is the amplitude of the magnetic field, ω is the angular frequency, β is the phase constant, z is the distance along the direction of propagation, and φ is the phase shift.

The amplitude of the magnetic field can be calculated from the electric field (E) using the formula:

H0 = E / Z

The angular frequency can be calculated from the operating frequency (f) using the formula:

ω = 2πf

The phase constant can be calculated from the wavelength (λ) using the formula:

β = 2π / λ

The phase shift (φ) is determined by the initial conditions of the wave. For a wave that is incident on a boundary, the phase shift is typically zero.

Please note that the actual calculations would require the use of a scientific calculator.