Question

In a double-slit arrangement the slits are separated by a distance equal to 100 times the wavelength of the light passing through the slits. (a)What is the angular separation in radians between the central maximum and an adjacent maximum? (b) What is the distance between these maxima on a screen 50 cm from the slits?

Short answer

a. The angular separation between the central maximum and adjacent maximum is 0.01 rad.

b. The distance between central axis and first maximum is 0.05 cm.

Step-by-step solution

Write the given data from the question

The separation distance between double slit arrangement is the 100 times the light passes through the slits,$d=100\lambda$.

Determine the formulas to calculate the angular separation and distance between the central axis and first minima:

The condition for the maxima in Young’s experiment is given as follows.$\mathit{d}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathbf{=}\mathit{m}\mathit{\lambda }$ …… (1)

Here, $\mathit{\lambda }$. is the wavelength, mis the order and$\mathit{\theta }$is the angular separation.

The expression to calculate the distance between the central axis and first minima is given as follows.

$\mathit{y}\mathbf{=}\mathit{L}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }$..…. (2)

Here,$\mathit{L}$ is the distance between the slit and screen.

Calculate the angular separation:

a.

Calculate the angular separation in radians between the central maximum and an adjacent maximum as below.

Substitute 1 for m and $100\lambda$ for d into equation (1).

$$\begin{gathered} \left(100\lambda \right)\sin \theta =1\times \lambda \\ \sin \theta =\frac{\lambda }{100\lambda } \end{gathered}$$

$$\begin{gathered} \theta ={\sin }^{-1}\left(\frac{1}{100}\right) \\ =0.572° \end{gathered}$$

Convert the angular separation from degrees to radian.

$$\begin{gathered} \theta =0.572\times \frac{\pi }{180} \\ =0.01\text{rad} \end{gathered}$$

Hence the angular separation between the central maximum and adjacent maximum is 0.01 rad.

Calculate the distance between the central axis and first maximum

b.

The distance between slit and screen, $L=50\text{cm}$

Calculate the distance between central axis and first maximum.

Substitute 50 cm for L and $0.{572}^0$for $\theta$into equation (2)

$$\begin{gathered} y=50\times {10}^{-2}\sin \left(0.572°\right) \\ =0.50\times 0.01 \\ =0.50\text{cm} \end{gathered}$$

Hence, the distance between central axis and first maximum is 0.50 cm.