The diffraction pattern due to a single slit obtained on a screen is shown in
Angular width of a central maximum :
It is defined as the angle between the directions of the first minima on two sides of the central maximum. That is, angular width of central maximum is $2\theta$.
The direction of the first minima on either side of the central maximum is given by
$\theta = \frac{\lambda}{d} \quad \cdots (i) $
which is called half angular width of central maximum.
Therefore, angular width of central maximum $= 2\theta = \frac{2\lambda}{d} \quad \cdots (ii)$
Linear width of central maximum :
Let $D$ be the distance of the screen from the centre $C$ of the slit.
The linear distance of the first minima from the centre $O$ of the screen is given by
$\because \text{ Arc } = \text{ angle } \times \text{ radius}$
$x = \theta D \quad $
$x = \frac{\lambda}{d} D = \frac{\lambda D}{d} \quad [\text{Using eqn. } (i)] $
The width of central maximum is equal to the linear distance between first minima on either side of central maximum. Therefore, width of central maximum, $\beta_0 = 2x$
$ \beta_0 = \frac{2\lambda D}{d} \quad \cdots (iii) $
Angular width of secondary maximum :
Angular width of $m^{th}$ secondary maximum is defined as the angular separation between the directions of $m^{th}$ and $(m+1)^{th}$ minima.
Therefore, angular width of $m^{th}$ secondary maximum,
$\phi = \theta_{m+1} - \theta_{m}$
Where , $\theta_m = \frac{m \lambda}{d}$ is the direction of $m^{th}$ minimum
$\theta_{m+1} = \frac{(m+1) \lambda}{d}$ is the direction of $(m+1)^{th}$ minimum
$ \therefore$ , $\phi = \frac{(m+1) \lambda}{d} - \frac{m \lambda}{d}$
$ \phi = \frac{\lambda}{d}$
Linear width of secondary maximum.
$\beta = \phi D$
$\beta = \frac{\lambda D}{d}$
Comparing eqs. (iii) and (iv), we get
$\beta_0 = 2 \beta$
Thus, width of central maximum is twice the width of secondary maximum.
Effect of width of slit on diffraction pattern :
Width of central maximum and secondary maxima is inversely proportional to the width ($d$) of slit. If width of slit increases, the width of central maximum and secondary maxima decreases. When width of slit is sufficiently large, secondary maxima of diffraction pattern disappear and the central maximum becomes a sharp point, which is the sharp image of the slit.
Thus, distinct diffraction pattern can be observed only if the width of slit is small or the slit is very narrow.
Factors on which width of central maximum Depends :
The width of the central maximum is directly proportional to:
(i) The wavelength of the light used, i.e., width of central maximum $\propto \lambda$. Therefore, width of central maximum is small for violet colour and large for red colour.
(ii) The distance $D$ between plane of slit and screen, so the width will increase with increase in $D$.
(iii) The width of central maximum is inversely proportional to the width ($d$) of the slit. If the width of the slit is small, width of central maximum is large and vice-versa.