# Double Slit Intensity Pattern

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```					 PHY3063                                                                                                          R. D. Field

Double Slit Intensity Pattern
Double Slit                             We form the superposition of the two
waves at the point P on the screen as
r1
follows.
θ
S1
d                         r2
Ψ1 = Aei ( kr1 −ωt )
Ψ2 = Aei ( kr2 −ωt )
S2                    θ

∆r =d sinθ                                                          Ψtot = Ψ1 + Ψ2
and thus

Ψtot = Ψ1 + Ψ2 = Aei ( kr1−ωt ) + Aei ( kr2 −ωt )
(             )                             (
= Aei ( kr1−ωt ) 1 + eik∆r = Aei ( kr1−ωt ) eik∆r / 2 eik∆r / 2 + e−ik∆r / 2                    )
= Aei ( kr1−ωt ) eik∆r / 2 2 cos(1 k∆r )
2
where k = 2π/λ. The path difference (for L >> d) is given by
∆r = r2 − r1 = d sinθ
The distance from the center of the two slits and the point P on the screen is
given by r = r1 + ∆r/2 and hence
Ψtot (r,θ ) = Aei ( kr−ωt ) 2 cos(πd sinθ / λ ) .
The intensity is proportional to the amplitude squared and hence
I (θ ) = 4I 0 cos2 (πd sinθ / λ )
where I0 is the intensity of a single wave (wave 1 or wave 2).
Double Slit                          5/2

2
P 1/2

r                              1

1/2
S1
d sinθ/λ

θ
d                                                                0
S2                                      L                            -1/2

-1

-3/2
Central Bright Spot
-2

Intensity

Department of Physics                                Chapter5_6.doc                                     University of Florida

```
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