Light with λ = 530 nm passes through a single slit and then illuminates a screen 9.0 m away. If the distance on the screen from the first dark fringe to the center of the interference pattern is 3.20 mm, what is the width of the slit?

tanα=x/L=3.2•10⁻³/9=3.56•10⁻⁴

tanα≈sinα
bsinα=kλ
b= kλ/sinα=1•530•10⁻⁹/3.56•10⁻⁴=1.5•10⁻³ m

To find the width of the slit, we can use the concept of diffraction and the formula for the angular position of the first dark fringe:

sinθ = λ / (wavelength) = λ / (a * sinθ)

where:
- θ is the angular position of the first dark fringe,
- λ is the wavelength of light,
- a is the width of the slit.

First, we need to find the value of θ. We can use the small-angle approximation since θ is small:

θ = tanθ ≈ 3.20 mm / 9.0 m

Now, we can substitute the values into the formula to find the width of the slit:

a = λ / (sinθ)

a = λ / (sin(tan^(-1)(3.20 mm / 9.0 m)))

Next, we need to convert the given wavelength from nm to meters:

λ = 530 nm = 530 * 10^(-9) m

Now we can substitute this value into the formula:

a = (530 * 10^(-9) m) / (sin(tan^(-1)(3.20 mm / 9.0 m)))

Calculating this expression will give us the width of the slit in meters.