A) \[{{\lambda }_{2}}\] is longer than \[{{\lambda }_{1}}\] and the ratio of the longer to the shorter wavelength is \[1.5\]
B) \[{{\lambda }_{1}}\] is longer than \[{{\lambda }_{2}}\] and the ratio of the longer to the shorter wavelength is 1.5
C) \[{{\lambda }_{1}}\] and \[{{\lambda }_{2}}\] are equal and their ratio is \[1.0\]
D) \[{{\lambda }_{1}}\] is longer than \[{{\lambda }_{2}}\] and the ratio of the longer to the shorter wavelength is \[2.5\]
Correct Answer: C
Solution :
The equation of nth principle maxima for wavelength \[\lambda \] is given by \[(a+b)\,\sin \theta =n\lambda \] where, a is the width of transparent portion and b is that of opaque portion. The width (a + b) is called the grating element. The spectral lines will overlap, i.e. they will have the same angle of diffraction of \[{{\lambda }_{1}}={{\lambda }_{2}}\] When a line of wavelength \[{{\lambda }_{1}}\] in order \[{{n}_{1}}\] coincides with a line of unknown wavelength \[{{\lambda }_{2}}\] in order \[{{n}_{2}}\] then \[{{n}_{2}}{{\lambda }_{2}}\,={{n}_{1}}{{\lambda }_{1}}\] or \[\frac{{{\lambda }_{1}}}{{{\lambda }_{1}}}=\frac{{{n}_{2}}}{{{n}_{1}}}\]
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