Question

In a double slit experiment to find the separation between slits by displacement method, the separation of images of slits were formed to be $16mm$ and $9mm$ respectively. The actual separation between slits will be:

Answer 1

Hint:When displacement method is used, the actual separation between slits can be found out by either using the separation between the images or by finding the focal length and displacement of the lens. Here we are only concerned with the separation between the images and not the focal length or displacement of the lens.

Formula used: The actual separation between two slits is given by the following expression:
\[d = \sqrt {{d_1} \times {d_2}} \]
Here, $d$ is the actual separation between slits ${S_1}$and ${S_2}$
${d_1}$is the separation between the two bright and enlarged images of the slits when a concave lens is kept closer to the slits and images are focussed as well.
${d_2}$ is the separation between the images when the convex lens is moved towards the screen and diminished, small and focussed images of the slits are formed in the screen.

Complete Step by step answer:
We first consider the magnification of the lens to be m. This magnification will be the same for both the convex lenses. Now, we know that the magnification produced in the lens is equal to the ratio between the separations between the two bright and enlarged images of the slits when a convex lens is kept closer to the slits and images are focussed as well to the separation between the two slits. Thus,
$m = \dfrac{{{d_1}}}{d}$
Magnification is also equal to the product of the actual separation between the two slits and the separation between the images when the convex lens is moved towards the screen. That is:
$m = d \times {d_2}$
Since both of these expressions are equal to m thus we can equate both these equations. Thus:
$d \times {d_2} = \dfrac{{{d_1}}}{d}$
$ \Rightarrow \dfrac{{{d_1}}}{{{d_2}}} = {d^2}$
Now, since in this question we are interested in finding the actual separation between the two slits, thus we can make d the subject of the formula. So,
$d = \sqrt {{d_1} \times {d_2}} $
From this question, ${d_1} = 16mm$and ${d_2} = 9mm$. Putting these values in the above equation, we get:
$\
  d = \sqrt {16 \times 9} = \sqrt {144} \\
   \Rightarrow d = 12mm \\
\ $

Hence, we find that the actual separation between the slits is equal to $12mm$.

Note:Since the separation between the two slits is also small (in millimetres) thus it is not fruitful in converting the units to their SI units (i.e. centimetres). The units should only be converted from one unit to another only when it directly hampers our solution. In any other case this process can be avoided.
The diffraction pattern of two slits of width D that are separated by a distance d is the interference pattern of two point sources separated by d multiplied by the diffraction pattern of a slit of width D.