A monochromatic parallel beam of light of wavelength $\lambda$ is incident normally on the plane containing slits  $S_{1}and{S}_2$. The slits are of unequal width such that intensity only due to one slit on screen is four times that only due to the other silt. The screen is placed perpendicular to x- axis as shown. The distance between slits is d and that between screen and slit is D. Match the statements in column-I with results in column –II.  ($\left(S_1{S}_2<<D and \lambda <<S_1{S}_2\right)$

	COLUMN-I		COLUMN-II
I)	The distance between two points on  screen having equal intensities such that intensity at those points is $\frac{1}{9}th$  of  maximum in intensity	P)	$\frac{D\lambda }{3d}$
II)	The distance between two points on screen having equal intensities, such that intensity at those points is $\frac{3}{9}th$ of maximum intensity.	Q)	$\frac{D\lambda }{d}$
III)	The distance between two points on screen having equal intensities, such that intensity at those points is $\frac{5}{9}th$ of maximum intensity	R)	$\frac{2D\lambda }{d}$
IV)	The distance between two points on screen having equal intensities, such that intensity at those points is  $\frac{7}{9}th$ of maximum intensity	S)	$\frac{3D\lambda }{d}$

 

The correct is B A- q,r,s B- p,q,r,s,t C- q,r,s D- p,q,r,s,t  
Resultant intensity I=I1 +I2+ $2\sqrt{I_1{I}_2\cos \varphi }$
Given  $I_1=4I_2$
Let $I_2=I_0,I_1=4I_0$ 

 $I=39I_{\max }\Rightarrow I_0=5I_0+4I_0\cos \varphi \Rightarrow \cos \varphi =1\varphi =\pi ,3\pi ,5\pi$. Distance of point from central maxima on either sided $y=Dd\lambda 2\pi \Delta \varphi =\lambda D2d,3\lambda D2d,5\lambda D2d,7\lambda D2d$ The distance between two points having equal intensities, such that intensity at those points is (19)th of maximum intensity  $\Rightarrow \Delta y=\lambda Dd,2\lambda Dd,3\lambda Dd,4\lambda Dd\mathrm{....}$
Hence q,r,s  

$I=39I_{\max }\Rightarrow 3I_0=5I_0+4I_0\cos \varphi \Rightarrow 12\varphi =2\pi 3,4\pi 3,8\pi 3,14\pi 3,16\pi 3$ . Distance of point from central maxima on either side $y=Dd\lambda \pi \Delta \varphi =\lambda D3d,2\lambda D3d,4\lambda D3d,5\lambda D3d,7\lambda D3d,8\lambda D3d$  The distance between two points having equal intensities, such that intensity at those points is (39)th of maximum
Intensity $\Rightarrow \Delta y=\lambda D3d,2\lambda D3d,\lambda Dd,5\lambda D3d,2\lambda Dd,8\lambda D3d,3\lambda Dd,10\lambda D3d$  
Hence p,q,r,s,t.

$I=5\mathrm{Im}ax9=5I_0\Rightarrow \cos \varphi =0\Rightarrow \theta \pi 2,3\pi 2,5\pi 2,7\pi 2,9\pi 2,11\pi \mathrm{2....}$ Distance of point from central maxima on either sides  $y=Dd\lambda 2\pi \Delta \varphi =\lambda D4d,5\lambda D4d,7\lambda D4d,9\lambda D4d,11\lambda D4d$ The distance between two points having equal intensities, such that intensity at those points is (59) th of maximum intensity  $\Rightarrow \Delta y=\lambda D2d,\lambda Dd,3\lambda D2d,2\lambda Dd,5\lambda D2d,3\lambda Dd\mathrm{...}$
Hence q,r,s. 

$I=79Imax=7I0\Rightarrow \cos \varphi =12\Rightarrow \varphi =3,5\pi 3,7\pi 3,11\pi 3,\mathrm{..}$ Distance of point from central maxima on either sides $y=Dd\lambda 2\pi \Delta \varphi =D6d,5\lambda D6d,7\lambda D6d,11\lambda D6d\mathrm{......}$ .The distance between two points having equal intensities, such that intensity at those points is (79) th of maximum 
Intensity.  $\Delta y=\lambda D3d,2\lambda D3d,\lambda Dd,4Dd,5\lambda 3D,2\lambda Dd,7\lambda D3d,8\lambda D3d,3\lambda Dd$
Hence p,q,r,s,t