# Show that the distribution of intensity in a double-slit pattern is given by Equation 36.9. Begin by assuming that the total magnitude of the electric field at point P on the screen in Figure 36.4 is the superposition of two waves, with electric field magnitudes E 1 = E 0 sin ω t E 2 = E 0 sin ( ω t + ϕ ) The phase angle in ϕ in E 2 is due to the extra path length traveled by the lower beam in Figure 36.4. Recall from Equation 33.27 that the intensity of light is proportional to the square of the amplitude of the electric field. In addition, the apparent intensity of the pattern is the time-averaged intensity of the electromagnetic wave. You will need to evaluate the integral of the square of the sine function over one period. Refer to Figure 32.5 for an easy way to perform this evaluation. You will also need the trigonometric identity sin A + sin B = 2 sin ( A + B 2 ) cos ( A − B 2 )

### Physics for Scientists and Enginee...

10th Edition
Raymond A. Serway + 1 other
Publisher: Cengage Learning
ISBN: 9781337553278

### Physics for Scientists and Enginee...

10th Edition
Raymond A. Serway + 1 other
Publisher: Cengage Learning
ISBN: 9781337553278

#### Solutions

Chapter
Section
Chapter 36, Problem 16P
Textbook Problem
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## Show that the distribution of intensity in a double-slit pattern is given by Equation 36.9. Begin by assuming that the total magnitude of the electric field at point P on the screen in Figure 36.4 is the superposition of two waves, with electric field magnitudes E 1 = E 0 sin ω t   E 2 = E 0 sin ( ω t + ϕ ) The phase angle in ϕ in E2 is due to the extra path length traveled by the lower beam in Figure 36.4. Recall from Equation 33.27 that the intensity of light is proportional to the square of the amplitude of the electric field. In addition, the apparent intensity of the pattern is the time-averaged intensity of the electromagnetic wave. You will need to evaluate the integral of the square of the sine function over one period. Refer to Figure 32.5 for an easy way to perform this evaluation. You will also need the trigonometric identity sin A + sin B = 2 sin ( A + B 2 ) cos ( A − B 2 )

Expert Solution
To determine

To show: The equation the intensity at a point in a double slit interference pattern is, I=Imaxcos2(πdsinθλ)

### Explanation of Solution

Given info: The equation of the magnitude of the electric fields is E1=E0sinωt and E2=E0sin(ωt+ϕ)

The formula to calculate the resultant of the electric field is,

E=E1+E2

Here,

E1 is the magnitude of the electric field.

E2 is the magnitude of the electric field.

Substitute E0sinωt for E1 and E0sin(ωt+ϕ) for E2 in above equation to find the value of E .

E=E0sinωt+E0sin(ωt+ϕ)=2E0sin(2ωt+φ2)cos(ϕ2)

For the average time the value of sin(2ωt+φ2) is 12 .

Substitute 12 for sin(2ωt+φ2) in above equation to find the value of E .

E=2E0(12)cos(ϕ2)=E0cos(ϕ2)

The formula to calculate the intensity is,

IE2

Substitute E0cos(ϕ2) for E in above equation to find the value of I

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