PHYSICS FOR SCI. & ENGR(LL W/WEBASSIGN)
PHYSICS FOR SCI. & ENGR(LL W/WEBASSIGN)
10th Edition
ISBN: 9781337888721
Author: SERWAY
Publisher: CENGAGE L
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Chapter 36, Problem 18P

Monochromatic coherent light of amplitude E0 and angular frequency ω passes through three parallel slits, each separated by a distance d from its neighbor. (a) Show that the time-averaged intensity as a function of the angle θ is

I ( θ ) = I max [ 1 + 2 cos ( 2 π d sin θ λ ) ] 2

(b) Explain how this expression describes both the primary and the secondary maxima. (c) Determine the ratio of the intensities of the primary and secondary maxima. Hint: See Problem 16.

(a)

Expert Solution
Check Mark
To determine

To show: The time averaged intensity as a function of angle θ is I(θ)=Imax[1+2cos(2πdsinθλ)]2 .

Answer to Problem 18P

The time averaged intensity as a function of angle θ is I(θ)=Imax[1+2cos(2πdsinθλ)]2 .

Explanation of Solution

Given info: The amplitude of monochromatic light is E0 and angular frequency is ω .

The amplitude of the monochromatic light is,

E1=E0sinωtE2=E0(sinωt+ϕ)E3=E0(sinωt+2ϕ)

Here,

ω is the angular frequency.

ϕ is the phase difference whose value is 2πdsin(θ)λ .

The total amplitude of monochromatic light is,

E=E1+E2+E3

Substitute E0sinωt for E1 , E0(sinωt+ϕ) for E2 , E0(sinωt+2ϕ) for E3 in the above formula as,

E=E0sinωt+E0(sinωt+ϕ)+E0(sinωt+2ϕ)

Apply the trigonometric identity to the above expression as,

E=2E0cos(ϕ2)sin(2ωt+ϕ2) +

Apply the trigonometric identity to the above expression as,

E=2E0cos(ϕ2)sin(2ωt+ϕ2)+E0sin(ωt+ϕ)

Add the above result to E2 to get the resultant field as,

E=2E0cos(ϕ2)sin(2ωt+ϕ2)+E0sin(ωt+ϕ)

As it is known that intensity of monochromatic light is directly proportional to the the square of the electric field that is,

IE2

Here,

I is the intensity.

E is the electric field.

The resultant field is obtained by square the above expression as,

E2=[[2E0cos(ϕ2)sin(2ωt+ϕ2)]+[E0sin(ωt+ϕ)]]2AB=4E20cos2(ϕ2)sin2(2ωt+ϕ2)+E02sin2(ωt+ϕ)+4E0cos(ϕ2)sin(2ωt+ϕ2)E0sin(ωt+ϕ)=E20(1+2cosϕ)2

Substitute 2πdsin(θ)λ for ϕ in the above expression as,

I(θ)=Imax[1+2cos(2πdsinθλ)]2

Conclusion:

Thus, the time averaged intensity as a function of angle θ is I(θ)=Imax[1+2cos(2πdsinθλ)]2 .

(b)

Expert Solution
Check Mark
To determine
The way in which the expression describes both the primary and secondary maxima.

Answer to Problem 18P

The expression describes both the primary and secondary maxima.

Explanation of Solution

Given info: The amplitude of monochromatic light is E0 and angular frequency is ω .

From the above expression obtained in part (a), the minimum interference is obtained when cosϕ is 12 and maximum interference is obtained when cosϕ is 1.

Conclusion:

Thus, the expression describes both the primary and secondary maxima.

(c)

Expert Solution
Check Mark
To determine
The ratio of the intensities of the primary and secondary maxima.

Answer to Problem 18P

The ratio of the intensities of the primary and secondary maxima is 19

Explanation of Solution

Given info: The amplitude of monochromatic light is E0 and angular frequency is ω .

Consider the amplitude of the monochromatic light is,

E1=E0sinωtE2=E0(sinωt+ϕ)E3=E0(sinωt+2ϕ)

The total amplitude of monochromatic light is,

E=E1+E2+E3

Substitute E0sinωt for E1 , E0(sinωt+ϕ) for E2 , E0(sinωt+2ϕ) for E3 in the above formula as,

E=E0sinωt+E0(sinωt+ϕ)+E0(sinωt+2ϕ)

Apply the trigonometric identity to the above expression as,

E=2E0cos(ϕ2)sin(2ωt+ϕ2) +

Apply the trigonometric identity to the above expression as,

E=2E0cos(ϕ2)sin(2ωt+ϕ2)+E0sin(ωt+ϕ)

Add the above result to E2 to get the resultant field as,

E=2E0cos(ϕ2)sin(2ωt+ϕ2)+E0sin(ωt+ϕ)

As it is known that intensity of monochromatic light is directly proportional to the  square of the electric field that is,

IE2

Here,

I is the intensity.

E is the electric field.

The resultant field is obtained by square the above expression as,

E2=[[2E0cos(ϕ2)sin(2ωt+ϕ2)]+[E0sin(ωt+ϕ)]]2=4E20cos2(ϕ2)sin2(2ωt+ϕ2)+E02sin2(ωt+ϕ)+4E0cos(ϕ2)sin(2ωt+ϕ2)E0sin(ωt+ϕ)=E20(1+2cosϕ)2

Substitute 2πdsin(θ)λ for ϕ in the above expression as,

I(θ)=Imax[1+2cos(2πdsinθλ)]2

The expression for the intensity for primary maxima is,

I(θ)=Imax[1+2cos(2πdsinθλ)]2

The expression for the intensity for secondary maxima is,

I(θ)=9Imax[1+2cos(2πdsinθλ)]2

Take the ratio of the above two expression as,

I(θ)priI(θ)sec=19

Conclusion:

Therefore, the ratio of the intensities of the primary and secondary maxima is 19 .

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Chapter 36 Solutions

PHYSICS FOR SCI. & ENGR(LL W/WEBASSIGN)

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