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(B) The resultant intensity $I$ for two coherent sources is given by $I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos \phi$. Since the radiators are identical,$I

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$2I_o$

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(B) The resultant intensity $I$ for two coherent sources is given by $I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos \phi$. Since the radiators are identical,$I_1 = I_2 = I_o$,so $I = 2I_o + 2I_o \cos \phi = 2I_o(1 + \cos \phi) = 4I_o \cos^2(\phi/2)$.Here,the total phase difference $\phi$ is the sum of the initial phase difference $\phi_i = \pi/4$ and the phase difference due to path difference $\Delta x = d \sin 	heta$.Path difference $\Delta x = d \sin 	heta = (\lambda/4) \sin 30^\circ = (\lambda/4) 	imes (1/2) = \lambda/8$.The phase difference due to path difference is $\Delta \phi = (2\pi/\lambda) \Delta x = (2\pi/\lambda) 	imes (\lambda/8) = \pi/4$.Total phase difference $\phi = \phi_i + \Delta \phi = \pi/4 + \pi/4 = \pi/2$.Substituting this into the intensity formula: $I = 4I_o \cos^2(\pi/4) = 4I_o 	imes (1/\sqrt{2})^2 = 4I_o 	imes (1/2) = 2I_o$.

Similar Questions

Two point sources separated by $d = 5\, \mu m$ emit light of wavelength $\lambda = 2\, \mu m$ in phase. $A$ circular wire of radius $R = 20\, \mu m$ is placed around the sources as shown in the figure. Determine the nature of interference at points $A, B, C,$ and $D$.

In the phenomenon of interference :-
$(a)$ Total energy is conserved
$(b)$ Total energy is not conserved
$(c)$ Only bright fringes are formed
$(d)$ The frequency of two sources is equal

In the interference of two coherent sources of different intensities,the ratio of maximum and minimum intensity is $\frac{I_{\max}}{I_{\min}} = \frac{144}{81}$. What is the ratio of their amplitudes?

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