JEE Main & Advanced Physics Wave Optics / तरंग प्रकाशिकी Useful Results

(1) Path difference : Path difference between the interfering waves meeting at a point P on the screen  is given by

\[\Delta ={{\Delta }_{i}}+{{\Delta }_{f}}\]; where  \[{{\Delta }_{i}}=\]initial path difference between the waves before the slits and \[{{\Delta }_{f}}=\]path difference between the waves after emerging from the slits. In this case \[{{\Delta }_{i}}=0\] (Commonly used condition). So \[\Delta ={{\Delta }_{f}}=\frac{xd}{D}=d\sin \theta \]

where \[x\] is the position of point P from central maxima.

For maxima at P :   \[\Delta =n\lambda \];  where \[n=0,\,\,\pm 1,\,\pm 2,\,...\]

and For minima at P : \[\Delta =\frac{(2n-1)\lambda }{2}\];  where \[n=\,\,\pm 1,\,\pm 2,\,...\]  

(2) Location of fringe : Position of nth bright fringe from central maxima \[{{x}_{n}}=\frac{n\lambda D}{d}=n\beta \]; \[n=0,\,1,\,2\,....\]

Position of nth dark fringe from central maxima

\[{{x}_{n}}=\frac{(2n-1)\,\lambda D}{2d}=\frac{(2n-1)\,\beta }{2}\]; \[n=1,\,2,3\,....\]

(3) Fringe width \[(\beta )\]:  The separation between any two consecutive bright or dark fringes is called fringe width. In YDSE all fringes are of equal width. Fringe width \[\beta =\frac{\lambda \,D}{d}\].

and angular fringe width \[\theta =\frac{\lambda }{d}=\frac{\beta }{D}\]

(4) In YDSE, if \[{{n}_{1}}\] fringes are visible in a field of view with light of wavelength \[{{\lambda }_{1}}\], while \[{{n}_{1}}\] with light of wavelength \[{{\lambda }_{2}}\] in the same field, then \[{{n}_{\mathbf{1}}}{{\lambda }_{\mathbf{1}}}={{n}_{\mathbf{2}}}{{\lambda }_{\mathbf{2}}}\].

(5) Separation \[(\Delta x)\] between fringes  

(i) Between nth bright and mth bright fringes \[(n>m)\] \[\Delta x=(n-m)\beta \]

(ii) Between nth bright and mth dark fringe

(a) If \[n>m\] then \[\Delta x=\left( n-m+\frac{1}{2} \right)\beta \]

(b) If \[n<m\] then \[\Delta x=\left( m-n-\frac{1}{2} \right)\beta \]

(6) Identification of central bright fringe : To identify central bright fringe, monochromatic light is replaced by white light. Due to overlapping central maxima will be white with red edges. On the other side of it we shall get a few coloured band and then uniform illumination.

If the whole YDSE set up is taken in another medium then \[\lambda \] changes so \[\beta \] changes

e.g. in water \[{{\lambda }_{w}}=\frac{{{\lambda }_{a}}}{{{\mu }_{w}}}\Rightarrow {{\beta }_{w}}=\frac{{{\beta }_{a}}}{{{\mu }_{w}}}=\frac{3}{4}{{\beta }_{a}}\]