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Wave Optics

Question
CBSEENPH12039788

In a diffraction pattern due to single slit of width a, the first minimum is obsrved at an angle 30o when light of wavelength 5000 space straight A with straight o on top is incident on the slit. The first secondary maximum is observed at an angle of:

  • sin-1 (2/3)

  • sin-1 (1/2)

  • sin-1 (3/4)

  • sin-1 (1/4)

Solution

C.

sin-1 (3/4)

Given that, first minimum is observed at an angle of 30o in a diffraction pattern due to a single slit of width a.
i.e., n =1, straight theta space equals space 30 to the power of straight o
According to the Bragg's law of diffraction,
space space space space space straight a space sin space straight theta space equals space nλ
rightwards double arrow space straight a space sin space 30 to the power of straight o space equals space left parenthesis 1 right parenthesis straight lambda space space left square bracket space straight n equals 1 right square bracket
rightwards double arrow space straight a space equals space 2 straight lambda space space space space space... bold space bold left parenthesis bold i bold right parenthesis bold space open curly brackets space sin space 30 to the power of straight o equals 1 half close curly brackets
For 1st secondary maxima,
rightwards double arrow space straight a space sinθ subscript 1 space equals space fraction numerator 3 straight lambda over denominator 2 end fraction
rightwards double arrow space sin space straight theta subscript 1 space equals space fraction numerator 3 straight lambda over denominator 2 straight a end fraction   ... (2)
Putting the value of a from Eqn. (i) to Eqn. (ii), we get
space space space space space space sin space theta subscript 1 space equals space fraction numerator 3 lambda over denominator 4 lambda end fraction
rightwards double arrow space sin space theta subscript 1 space equals space 3 over 4
rightwards double arrow space space theta subscript 1 italic space italic equals italic space space sin to the power of negative 1 end exponent open parentheses 3 over 4 close parentheses