In an interference pattern, the amplitudes of the interfering waves are indeed squared to determine the resulting intensity. When two or more waves overlap and interfere with each other, the resulting intensity at a given point is determined by the superposition of the amplitudes of the individual waves.
The principle of superposition states that when waves meet, their amplitudes add together to form a new wave. If the waves are in phase (crests and troughs align), they constructively interfere, leading to an increased amplitude and a brighter region in the interference pattern. Conversely, if the waves are out of phase (crest of one wave aligns with the trough of another), they destructively interfere, resulting in a decreased amplitude and a darker region in the interference pattern.
To determine the intensity of the resulting wave, the amplitudes of the interfering waves are squared. Intensity is proportional to the square of the amplitude, meaning that the intensity is directly related to the amplitude squared. This relationship allows us to calculate the brightness or darkness of the interference pattern.
If the amplitudes were not squared when calculating the interference pattern, it would not accurately represent the distribution of light intensity. Squaring the amplitudes is essential for capturing the constructive and destructive interference effects that occur when waves interact. By squaring the amplitudes, we account for the energy carried by the waves and obtain an accurate representation of the resulting intensity pattern.