Question

Light that has a wavelength of 530 nm is incident on a single slit that is 25.0 um wide. 1) Determine the angular location of the first dark fringe that is formed on a screen behind the slit. (Express your answer to three significant figures.) Submit 2) Determine the angular location of the second dark fringe that is formed on a screen behind the slit. (Express your answer to three significant figures.) Submit 3) Determine the angular location of the third dark fringe that is formed on a screen behind the slit. (Express your answer to three significant figures.) Submit

Answer 1

The liners position of nth dark fringe is given by

y_{​​​​​​n} = (2n-1)kD/2d

First dark fringe is given corresponding to n= 1

i.e y_{​​​​​​n} = (2n-1)kD/2d

Where D is the distance of screen from slit and k is the wavelength of light used, d is the slit width

Now, n = 1(for first dark fringe)

k = 530 nm = 530*10^-9 m

And d = 25.0 um = 25.0*10^-6 m

Using these values in eqn 1

y = (2*1-1)*530*10^-9*D/2*25.0*10^-6

= 530*10^-9D/50.0*10^-6

y = 10.36*10^-3*D

Now angular position is given by

theta = y/D

= 10.36*10^-3*D/D

Theta = 10.3*10^-3 rad

= 10.3*10^-3*180/π degrees

= 590.44*10^-3

Theta = 0.590°

2. Similarly, angular position for second dark fringe would correspond to n= 2

i.e. y = (2*2-1)* 530*10^-9*D/2*25.0*10^-6

^{₌ ​​​​​}​​​​​3*530*10^-9*D/2*25.0*10^-6

= 31.8*10^-3* D

Now, angular position

theta = y/D

= 31.8*10^-3*D/D

= 31.8*10^-3 rad

= 31.8*10^-3*180/π degrees

= 1822.92*10^-3

Theta = 1.82°

3. For n= 3,

y = (2*3-1)*530*10^-9*D/2*25.0*10^-6

= 5*530*10^-9*D/2*25.0*10^-6

= 53*10^-3*D

And theta = y/D

= 53*10^-3*D/D

= 53*10^-3 rad

= 53*10^-3*180/π degrees

= 3038.21*10^-3

Theta = 3.03°