question_answer

Two coherent sources S1 and S2 are separated by a distance four times the wavelength l of the source. The sources lie along y axis whereas a detector moves along + x axis. Leaving the origin and far off points the number of points where maxima are observed is

A)            2    

B)            3

C)            4    

D)            5

Correct Answer: B

Solution :

           From \[\Delta {{S}_{1}}{{S}_{2}}D,\] \[{{({{S}_{1}}D)}^{2}}={{({{S}_{1}}{{S}_{2}})}^{2}}+{{({{S}_{2}}D)}^{2}}\] \[{{({{S}_{1}}P+PD)}^{2}}={{({{S}_{1}}{{S}_{2}})}^{2}}+{{({{S}_{2}}D)}^{2}}\] Here \[{{S}_{1}}P\] is the path difference \[=n\lambda \] for maximum intensity. \[\therefore {{(n\lambda +{{x}_{n}})}^{2}}={{(4\lambda )}^{2}}+{{({{x}_{n}})}^{2}}\] or \[{{x}_{n}}=\frac{16{{\lambda }^{2}}-{{n}^{2}}{{\lambda }^{2}}}{2n\lambda }\] Then \[{{x}_{1}}=\frac{16{{\lambda }^{2}}-{{\lambda }^{2}}}{2\lambda }=7.5\,\lambda \] \[{{x}_{2}}=\frac{16{{\lambda }^{2}}-4{{\lambda }^{2}}}{4\lambda }=3\lambda \] \[{{x}_{3}}=\frac{16{{\lambda }^{2}}-9{{\lambda }^{2}}}{6\lambda }=\frac{7}{6}\lambda \] \[{{x}_{4}}=0\]. \[\therefore \]Number of points for maxima becomes 3.