Question
A plane wave of monochromatic light falls normally on a uniform thin layer of oil which covers a glass plate. The wavelength of source can be varies continuously. Complete destructive interference is observed for $$\lambda = 5000\,\mathop {\text{A}}\limits^ \circ $$ and $$\lambda = 1000\,\mathop {\text{A}}\limits^ \circ $$ and for no other wavelength in between. If $$\mu $$ of oil is $$1.3$$ and that of glass is $$1.5,$$ the thickness of the film will be
A.
$$6.738 \times {10^{ - 5}}cm$$
B.
$$5.7 \times {10^{ - 5}}cm$$
C.
$$4 \times {10^{ - 5}}cm$$
D.
$$2.8 \times {10^{ - 5}}cm$$
Answer :
$$6.738 \times {10^{ - 5}}cm$$
Solution :
In this case, both the rays suffere a phase change of $${180^ \circ }$$ and the conditions for destructive interference is
$$\eqalign{
& 2{\text{nd}} = \left( {m + \frac{1}{2}} \right){\lambda _1}\,\,{\text{and}}\,\,2{\text{nd}}\left( {m + \frac{3}{2}} \right){\lambda _2} \cr
& \therefore \frac{{m + \frac{1}{2}}}{{m + \frac{3}{2}}} = \frac{{{\lambda _2}}}{{{\lambda _1}}} = \frac{{5000}}{{700}} = \frac{5}{7} \cr
& {\text{and}}\,\,d = \frac{{\left( {m + \frac{1}{2}} \right){\lambda _1}}}{{2n}} = \frac{{2.5 \times 7000}}{{2 \times 1.3}} \cr
& = 6.738 \times {10^{ - 5}}cm \cr} $$