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Chapter 25: Problem 4

Sketch a sinusoidal wave with an amplitude of \(2 \mathrm{cm}\) and a wavelength of \(6 \mathrm{cm} .\) This wave represents the electric field portion of a visible EM wave traveling to the right with intensity \(I_{0}\). (a) Sketch an identical wave beneath the first. What is the amplitude (in centimeters) of the sum of these waves? (b) What is the intensity of the new wave? (c) Sketch two more coherent waves beneath the others, one of amplitude \(3 \mathrm{cm}\) and one of amplitude \(1 \mathrm{cm},\) so all four are in phase. What is the amplitude of the four waves added together? (d) What intensity results from adding the four waves?

Short Answer

Expert verified
Question: Given four coherent sinusoidal waves with amplitudes of 2 cm, 2 cm, 3 cm, and 1 cm, and a wavelength of 6 cm, find the intensity of the resulting wave when all four waves are added together. Answer: The intensity of the resulting wave when all four coherent sinusoidal waves are added together is 16I₀.

Step by step solution

01

Understanding the relationship between amplitude and intensity

The intensity (\(I\)) of a wave is directly proportional to the square of its amplitude (\(A\)). Mathematically, it is given by \(I \propto A^{2}\).
02

Amplitude of the sum of two identical sinusoidal waves

In part (a), we are asked to sketch two identical sinusoidal waves with amplitude 2 cm and wavelength 6 cm and find the amplitude of the sum of these two waves. Since they are identical and in phase, their amplitudes simply add up: \(A_{sum} = A_1 + A_2 = 2 + 2 = 4 \mathrm{cm}\).
03

Intensity of the new wave

In part (b), we find the intensity of the new wave formed after summing the two identical sinusoidal waves. Since intensity is proportional to the square of amplitude, the new intensity is: \(I_{new} = I_0 \left( \frac{A_{sum}}{A_1} \right)^2 = I_0 \left( \frac{4}{2} \right)^2 = 4 I_0\).
04

Adding more coherent waves

In part (c), we are adding two more coherent sinusoidal waves beneath the previous two waves, one with an amplitude of 3 cm and one with an amplitude of 1 cm, so that all waves are in phase. The total amplitude can be found as: \(A_{total} = A_{sum} + A_3 + A_4 = 4 + 3 + 1 = 8 \mathrm{cm}\).
05

Intensity of the resulting wave

In part (d), we need to find the intensity of the resulting wave when all four waves are added together. The resulting intensity can be calculated as: \(I_{result} = I_0 \left( \frac{A_{total}}{A_1} \right)^2 = I_0 \left( \frac{8}{2} \right)^2 = 16 I_0\).

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