Four coherent EM waves have intensities of lo, 0.30010, 0.200/0, and 0.400/o. The second is 180° out of phase with the first; the third and fourth are in phase with the first. What is the intensity of the superposition of the four? lo
Four coherent EM waves have intensities of lo, 0.30010, 0.200/0, and 0.400/o. The second is 180° out of phase with the first; the third and fourth are in phase with the first. What is the intensity of the superposition of the four? lo
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ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
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![### Problem Description
**Question:**
Four coherent EM waves have intensities of \( I_0, 0.300I_0, 0.200I_0, \) and \( 0.400I_0 \). The second is 180° out of phase with the first; the third and fourth are in phase with the first. What is the intensity of the superposition of the four?
[ ] \( I_0 \)
### Explanation
To solve this problem, we need to understand the impact of phase differences on the interference of electromagnetic waves. Here, we know:
- **Wave 1:** Intensity \( I_1 = I_0 \)
- **Wave 2:** Intensity \( I_2 = 0.300I_0 \) (180° out of phase with Wave 1)
- **Wave 3:** Intensity \( I_3 = 0.200I_0 \) (in phase with Wave 1)
- **Wave 4:** Intensity \( I_4 = 0.400I_0 \) (in phase with Wave 1)
### Calculation
1. Calculate the resultant amplitude of the superposition:
Since Wave 2 is 180° out of phase, its amplitude will subtract from that of Wave 1. Waves 3 and 4, in phase, will add their amplitudes to that of Wave 1.
2. Use the relationship between intensity and amplitude:
\[
I = kA^2
\]
Where \( I \) is the intensity, \( A \) is the amplitude, and \( k \) is a constant.
3. Determine the combined intensity using superposed amplitudes:
\[
A_{\text{total}} = \sqrt{A_1^2} - \sqrt{A_2^2} + \sqrt{A_3^2} + \sqrt{A_4^2}
\]
\[
I_{\text{total}} = k(A_{\text{total}})^2
\]
With this calculation, you can determine the final intensity of the superposition.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbeda2708-6e0a-400e-bdde-51711af4864b%2F980f8671-ced7-4ac6-8543-544e0c24b9c0%2F0z2k2bb_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Description
**Question:**
Four coherent EM waves have intensities of \( I_0, 0.300I_0, 0.200I_0, \) and \( 0.400I_0 \). The second is 180° out of phase with the first; the third and fourth are in phase with the first. What is the intensity of the superposition of the four?
[ ] \( I_0 \)
### Explanation
To solve this problem, we need to understand the impact of phase differences on the interference of electromagnetic waves. Here, we know:
- **Wave 1:** Intensity \( I_1 = I_0 \)
- **Wave 2:** Intensity \( I_2 = 0.300I_0 \) (180° out of phase with Wave 1)
- **Wave 3:** Intensity \( I_3 = 0.200I_0 \) (in phase with Wave 1)
- **Wave 4:** Intensity \( I_4 = 0.400I_0 \) (in phase with Wave 1)
### Calculation
1. Calculate the resultant amplitude of the superposition:
Since Wave 2 is 180° out of phase, its amplitude will subtract from that of Wave 1. Waves 3 and 4, in phase, will add their amplitudes to that of Wave 1.
2. Use the relationship between intensity and amplitude:
\[
I = kA^2
\]
Where \( I \) is the intensity, \( A \) is the amplitude, and \( k \) is a constant.
3. Determine the combined intensity using superposed amplitudes:
\[
A_{\text{total}} = \sqrt{A_1^2} - \sqrt{A_2^2} + \sqrt{A_3^2} + \sqrt{A_4^2}
\]
\[
I_{\text{total}} = k(A_{\text{total}})^2
\]
With this calculation, you can determine the final intensity of the superposition.
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