In a single slit diffraction pattern with slit width a and wavelength of light γ, find the angular position of first minima if screen distance is D(D >> a)

The correct option is γ/a.

To find the angular position of the first minimum in a single-slit diffraction pattern, let us analyze the path difference of light waves passing through the slit.

Consider a single slit of width $a$ illuminated by a parallel beam of light of wavelength $\gamma$. The screen is placed at a distance $D$ such that $D>>a$.

For a diffraction pattern produced by a single slit, the condition for minima (dark fringes) on the screen is given by the formula:
$a\sin \theta =n\gamma$
where:
• $a$ is the slit width,
• $\theta$ is the angular position (diffraction angle) of the minimum,
• $n$ is an integer representing the order of the minimum ($n=\pm 1,\pm 2,\pm 3,...$),
• $\gamma$ is the wavelength of light.

For the first minimum, we set $n=1$. This gives the relation:
$a\sin \theta =\gamma$

Solving for $\sin \theta$:
$\sin \theta =\frac{\gamma }{a}$

Since the screen distance $D$ is much larger than the slit width $a$ ($D>>a$), the angle of diffraction $\theta$ is very small. For very small angles expressed in radians, we can use the small-angle approximation:
$\sin \theta \approx \theta$

Substituting this approximation into the equation, we get the angular position of the first minimum:
$\theta =\frac{\gamma }{a}$