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An interference pattern is produced by light of wavelength from a distant source incident on two identical parallel slits separated by a distance (between centers) of 0.530 mm. (a) If the slits are very narrow, what would be the angular positions of the first-order and second-order, two-slit interference maxima? (b) Let the slits have width 0.320mm. In terms of the intensityI0 at the center of the central maximum, what is the intensity at each of the angular positions in part (a)?

Short Answer

Expert verified

(a) The angular positions of the first-order and second-order, two-slit interference maxima are0.0627 and0.1254 respectively.

(b) The intensity at each of the angular positions in part (a) is0.249I0 and 0.0256I0.

Step by step solution

01

Formula used to solve the question

Intensity is given by

I=I0(cos2ϕ2)(sinβ2β2)2

Where

ϕ=2πdsinθλ

And

β=2πasinθλ

02

Determine the angular positions of the first-order and second-order

Given:

λ=580nm=580*10-9md=0.530mm=0.530*10-3ma=0.320mm=0.320*10-3m

When the two slits are very narrow, the interference pattern will be affected only by the two-slits effect. And hence, there is no diffraction pattern due to the single-slit diffraction.

So, the angles of bright fringes of the double-slit interference is given by

sinθ=mλd

Hence, the angular position is given by

θ=sin-1(mλd) (1)

For first order and second order maximum, m=±1,±2.

Plug these values in (1),

θ1=sin1(1.05801090.530103)=0.0627θ2=sin1(2.05801090.530103)=0.1254

03

Determine the intensity at each of the angular positions

The intensity of four fringes including the diffraction effect is given by,

I=I0(cos2ϕ2)(sinβ2β2)2

Where

ϕ=2πdsinθλ

And

β=2πasinθλ

Therefore,

role="math" localid="1668176980220" I=I0(cos2πdsinθλ)(sinπasinθλπasinθλ)2

Plug the given values for first-order,

I1=I0(cos2πdλλd)(sinπaλλdπaλλd)2I1=I0(cos2π)(sinπadπad)2I1=I0(cos2π)(sinππ+0.320++1030.50+03π0.3201030.53003)2=0.249I0

Similarly for second-order,

I2=0.0256I0

Thus, the angular positions of the first-order and second-order, two-slit interference maxima are 0.0627and 0.1254respectively. The intensity at each of the angular positions in part (a) is0.249I0 and 0.0256I0.

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