Three lenses L_1, L_2, L_3 are placed co-axia | vedclass

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(C) Given: Focal lengths $f_1 = +30 \, cm$ (convex),$f_2 = -10 \, cm$ (concave),$f_3 = +5 \, cm$ (convex).Since the incident beam is parallel to the p

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$20$

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(C) Given: Focal lengths $f_1 = +30 \, cm$ (convex),$f_2 = -10 \, cm$ (concave),$f_3 = +5 \, cm$ (convex).Since the incident beam is parallel to the principal axis and the emergent beam from $L_3$ is also parallel to the principal axis (as it converges at the focus of $L_3$,which implies the rays incident on $L_3$ must be parallel to the principal axis),the combination of $L_1$ and $L_2$ must act as a system that produces a parallel beam from a parallel beam.For a system of two lenses separated by distance $d$ to act as a beam expander/collimator,the effective focal length of the combination must be infinite,or the image formed by $L_1$ must coincide with the object point for $L_2$.Using the formula for the equivalent focal length of two lenses separated by distance $d$:$\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$For the output to be parallel,$F = \infty$,so $\frac{1}{F} = 0$.$0 = \frac{1}{30} + \frac{1}{-10} - \frac{d}{30 	imes (-10)}$$0 = \frac{1}{30} - \frac{1}{10} + \frac{d}{300}$$\frac{1}{10} - \frac{1}{30} = \frac{d}{300}$$\frac{3 - 1}{30} = \frac{d}{300}$$\frac{2}{30} = \frac{d}{300}$$d = \frac{2 	imes 300}{30} = 20 \, cm$.

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