Question

In a single-slit diffraction pattern, the central fringe is 360 times as wide as the slit. The screen is 14,000 times farther from the slit than the slit is wide. What is the ratio /W, where is the wavelength of the light shining through the slit and W is the width of the slit

Answer 1

The ratio of λ / W in a single-slit diffraction pattern is 1 / 14,000.

Step-by-step explanation:

To calculate the width of the slit in a single-slit diffraction pattern, we can use the formula:

λ / W = 360

Given that the central fringe is 360 times as wide as the slit, we can substitute this value into the equation:

λ / W = 360

Since the screen is 14,000 times farther from the slit than the slit is wide, we can also use the formula:

D / W = 14,000

Substituting the given values into the equation:

D / W = 14,000

We can rearrange this equation to solve for the ratio:

λ / D = 1 / 14,000

Therefore, the ratio λ / W is equal to 1 / 14,000.

Answer 2

0.01286

Step-by-step explanation:

In a given single-slit, the central fringe (Y) is 360 times as wide as the slit (a). Then

2Y₁ = 360a

Y₁ = 360a/2

= 180a

The distance D = 14000a

In a given single-slit diffraction, the ratio =
`(\lambda )/(W)`

and since the angle is infinitesimally small;

sin θ ≅ tan θ =
`(Y)/(D)`

∴

For the first dark fringe;

Suppose:
`(a)/(2)sin \theta = (\lambda )/(2)`

then,

`(a)/(2) \ (Y_1)/(D) = (\lambda )/(2)`

`aY_1 = \lambda D`

`(\lambda )/(a) = (Y_1)/(D)\\ \\ \\ \implies (180\ a)/(14000 \ a) \\ \\ \mathbf{(\lambda )/(a) = 0.01286}`