Two parallel beams of light P and Q (separation d) containing radiations of wavelengts `4000Å` and 5000 Å (which are mutually coherent in each wavelength separately) are incident normally on a prism as shown in figure the refractive index of the prism as a function of wavelength is given by the relation `mu(lamda)=1.20+(b)/(lamda^(2))` where `lamda` is in Å and b is a positive constant. The value of b is such that the condition for total reflection at the face AC is just satisfied for one wavelength and is not satisfied for the other. A convergent lens is used to bring these transmitted beams into focus. If the intensities of the upper and the lower beams immediately after transmission from the face AC, are 4I and I respectively, find the resultant intensity at the focus.

Answer

9I

Explanation

Path difference between rays 1 and 2:
`Deltax=mu(QS)-PR` …(i)
Further `(QS)/(PS)=sini,(PR)/(PS)=sinr`
`therefore(PR//PS)/(QS//PS)=(sinr)/(sini)=muthereforemu(QS)=PR`
Substituting in equation (i), we get `Deltax=0`
`therefore` Phase difference between rays 1 and 2 will be 0 or these two rays will interfere constructively. So maximum intensity will be obtained from their interference or
`I_(max)=(sqrt(I_(1))+sqrt(I_(2)))^(2)=(sqrt(4I)+sqrt(I))^(2)=9I`