Solution :

Let a source S of monochrome light be placed at the focus of a convex lens $L_1$.

A parallel beam of light and lence a wavepoint gets incident on a slit AB of width d.

The incident wavefront disturbs all parts of the slit AB simultaneously .

According to Huygen's theory , all parts of the slit AB will become source of secondary wavelets spreads out it all direction , thereby causing diffraction of light after it emerges through slilt AB will become source of secondary wavelets , which all starts in the same phase .

These wavelength spread out in all direction , thereby causing diffraction of light after it emerge through slit AB .

The wave let from any two corresponding points of the two slits reach the point add constructively to produce a central bright fringe

(ii) Suppose the secondary wave lets diffected at an angle $\theta$ are focused at point P.

The secondary wavelets start ferom different parts of the slit in same phase but they reach the point P in different phases.

Draw perpendicular AN from A on to the ray B.

Then the path difference between the wavelets from A and B will be $P= BP - AP$

$\qquad= BN=AB \sin \theta=d \sin \theta$

let the point be so located on the screen that the path difference $p= \lambda$ and let the cycle $ \theta-\theta_1$

$\therefore d \sin \theta_1= \lambda$

we can divide the slit AB into two halves AC and CB

the path difference between A and C will be $\large\frac{\lambda}{2}$

The condition for first dark finge is $d \sin \theta_1= \lambda$

The condition for second dark finge is $d \sin \theta_2= 2 \lambda$

$d \sin \theta _n = n \lambda$

Suppose the point p is so located that $p= 3/2 \lambda$

Condition for first secondary maximum is $d \sin \theta _2^1= \large\frac{3 \lambda}{2}$

Condition for nth secondary maximum is $d \sin \theta _n^1= (2n+1)\large\frac{ \lambda}{2}$