(a) What are coherent sources of light? Two slits in Young's double-slit experiment are illuminated by two different sodium lamps emitting light of the same wavelength.
Why is no interference pattern observed?
(b) Obtain the condition for getting dark and bright fringes in Young's experiment. Hence write the expression for the fringe width.
(c) If s is the size of the source and its distance from the plane of the two slits, what should be the criteria for the interference fringes to be seen?

(a) Coherent sources:
Sources emitting waves of same frequency or wavelength having either zero or a constant phase difference are said to be coherent sources of light. 

Two independent sources of light do not fulfil the requirement of constant phase difference. Since, the sodium lamps considered here are independent sources of light and are not coherent sources we cannot observe interference pattern. 

(b) For bright fringes (maxima), 

$\mathrm{Path} \mathrm{difference}, \frac{\mathrm{xd}}{\mathrm{D}} = \mathrm{n\lambda }$
$\therefore$                        $\mathrm{x} = \mathrm{n}\frac{\mathrm{\lambda D}}{\mathrm{d}}$
where,  n = 0, 1, 2, 3... so on. 

For dark fringes (minima),

Path difference,   $\frac{\mathrm{xd}}{\mathrm{D}} = (2\mathrm{n}-1) \mathrm{\lambda }/2$
$\therefore$                           $\mathrm{x} = (2\mathrm{n}-1) \frac{\mathrm{\lambda }}{2}\frac{\mathrm{D}}{\mathrm{d}}\text{'}$
Where n = 0, 1, 2, 3 ...so on. 

The separation between the centre of two consecutive bright fringes is the width of a dark fringe. 

$\therefore$ Fringe width,    $\mathrm{\beta } = {\mathrm{x}}_{\mathrm{n}}-{\mathrm{x}}_{\mathrm{n}-1}$ 

                            $\mathrm{\beta } = \mathrm{n}\frac{\mathrm{\lambda D}}{\mathrm{d}}-(\mathrm{n}-1)\frac{\mathrm{\lambda D}}{\mathrm{d}}$ 

i.e.,                      $\mathrm{\beta } = \frac{\mathrm{\lambda D}}{\mathrm{d}}$ 

(c) The condition for interference fringes is to be:
$\frac{\mathrm{S}}{\mathrm{d}}<\mathrm{\lambda }/\mathrm{d}$.
where, 
s is the source and, 
d is the distance between the two slits.