Question

What will be the width of the slit if the distance between the first and third minima in the diffraction pattern is $3mm$? The distance between screen and single slit is $50cm$ which is illuminated with light of wavelength $6000\dot A$.
A) $0.1mm$
B) $0.2mm$
C) $0.3mm$
D) $0.4mm$

Answer 1

-Hint:- Convert all the given values to their S.I unit.
Now, evaluate the distance between the first and third minima.
Next, we can now calculate the width of slit by the expression –
$x = \dfrac{{\lambda D}}{d}$
where, $x$ is the distance between first and third minima,
$\lambda $ is the wavelength of light,
$D$ is the distance between screen and slit and
$d$ is the width of slit.

Complete Step by Step Solution:-
Let the distance between first and third minima be $x$ and first minima be ${x_1}$ and third minima be ${x_3}$. Therefore,
$
  x = {x_3} - {x_1} \\
  x = 3mm \\
  x = 3 \times {10^{ - 3}}m \\
 $
Let the distance between the screen and single slit be $D$. So, according to the question it is given that –
$
  D = 50cm \\
  D = 0.5m \\
 $
Let the width of slit be $d$ and wavelength of light be $\lambda $.
According to the question, it is given that
$
  \lambda = 6000\dot A \\
  \lambda = 6000 \times {10^{ - 10}}m \\
 $
The position of ${n^{th}}$ minima in the diffraction pattern is given by –
${x_n} = n\dfrac{{D\lambda }}{d}$
The above can also be rewritten by transposition as –
$d = n\dfrac{{D\lambda }}{x} \cdots (1)$
Putting the values of $x,n,D$ and $\lambda $ in their respective places in equation $(1)$
$
  d = (3 - 1)\dfrac{{0.5 \times 6000 \times {{10}^{ - 10}}}}{{3 \times {{10}^{ - 3}}}} \\
  d = \dfrac{{2 \times 0.5 \times 6 \times {{10}^3} \times {{10}^{ - 10}}}}{{3 \times {{10}^{ - 3}}}} \\
 $
By further solving, we get
$
  d = \dfrac{{6 \times {{10}^{ - 4}}}}{3} \\
  d = 2 \times {10^{ - 4}}m \\
 $
Now, converting metre to millimetre, we get
$d = 0.2mm$

Therefore, the correct answer is option (B).

Note:- Diffraction is a specialized case of scattering of light in which an object with regularly repeating features produces an orderly diffraction of light in a diffraction pattern. The diffraction plays an important role in limiting the resolving power of any optical instrument.