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Statement for Linked Answer Questions 52 and 53: A monochromatic plane wave of wavelength \(60\pi\) mm is propagating in the direction as shown in the figure below. \(E_i\), \(E_r\), and \(E_t\) denote incident, reflected, and transmitted electric field vectors associated with the wave. The angle of incidence \(\theta_i\) and the expression for \(E_i\) are

1. 60° and E0/√2 (aₓ + a_z) e^(-j10⁴(x+z)/(3√2)) V/m
2. 45° and E0/√2 (aₓ + a_z) e^(-j10⁴(x+z)/(3√2)) V/m
3. 45° and E0/√2 (aₓ + a_z) e^(-j10⁴(x+z)/3) V/m
4. 60° and E0/√2 (aₓ + a_z) e^(-j10⁴(x+z)/3) V/m

Correct answer: 45° and E0/√2 (aₓ + a_z) e^(-j10⁴(x+z)/(3√2)) V/m

Solution

The direction shown corresponds to equal components along the relevant axes, giving an incidence angle of 45°. The phase factor follows from \(\beta=2\pi/\lambda\) with \(\lambda=60\pi\) mm, leading to the stated exponential form.

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