Question

Two slits of width 2 μm, each in an opaque material, are separated by a center-to-center distance of 6 μm. A monochromatic light of wavelength 450 nm is incident on the double-slit. One finds a combined interference and diffraction pattern on the screen.

(a) How many peaks of the interference will be observed in the central maximum of the diffraction pattern?

(b) How many peaks of the interference will be observed if the slit width is doubled while keeping the distance between the slits same?

(c) How many peaks of interference will be observed if the slits are separated by twice the distance, that is, 12 μm,

while keeping the widths of the slits same?

(d) What will happen in (a) if instead of 450-nm light another light of wavelength 680 nm is used?

(e) What is the value of the ratio of the intensity of the central peak to the intensity of the next bright peak in (a)?

(f) Does this ratio depend on the wavelength of the light?

(g) Does this ratio depend on the width or separation of the slits?

Answer 1

given

Two slits of width 2 μm

= monochromatic light of wavelength 450 nm

= 450 x 10^-9 m

a )

we have equation

D₁ sin₁ = n

₁ = n / D₁

₁ = 1 x 450 x 10^-9 / 2 x 10^-6

= 0.225 rad

m₁ = d₁₁ /

= 6 x 10^-6 x 0.225 / 450 x 10^-9

m₁ = 3 rad

b )

₂ = / 2 D₁

= 450 x 10^-9 / 2 x 2 x 10^-6

₂ = 0.112 rad

m₂ = d₂₂ /

= 6 x 10^-6 x 0.1125 / 450 x 10^-9

m₂ = 1.5 rad

c )

m₃ = d₂₁ /

= 2 x 6 x 10^-6 x 0.225 / 450 x 10^-9

m₃ = 6 rad

d )

₄ = ' / D₁

= 680 x 10^-9 / 2 x 10^-6

= 0.34 rad

m₄ = d₁₄ /

= 6 x 10^-6x 0.34 / 680 x 10^-9

m₄ = 3 rad

e )

using equation

I = I_o ( sin/ )²

Io / I = ( / sin )²

Io / I = ( ( m x 3.14 x D₁ / d₁ ) / sin( m x 3.14 x D₁ / d₁ ) )²

= ( ( 1 x 3.14 x 2 x 10^-6 / 6 x 10^-6 ) / sin( 1 x 3.14 x 2 x 10^-6 / 6 x 10^-6 ) )²

Io / I = 1.46

f )

using equation

I = I_o ( sin/ )²

Io / I = ( / sin )²

= ((m D / d) / sin(m D / d) )²

g )

it is also same

Io / I = ((m D / d) / sin(m D / d) )