The ratio of the dispersive powers of two len | vedclass

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(D) For an achromatic combination of two thin lenses in contact,the condition is $\frac{\omega_1}{f_1} + \frac{\omega_2}{f_2} = 0$,which implies $\fra

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$15 \, cm$ and $-20 \, cm$

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(D) For an achromatic combination of two thin lenses in contact,the condition is $\frac{\omega_1}{f_1} + \frac{\omega_2}{f_2} = 0$,which implies $\frac{\omega_1}{\omega_2} = -\frac{f_1}{f_2}$.Given $\frac{\omega_1}{\omega_2} = \frac{4}{3}$,we have $\frac{f_1}{f_2} = -\frac{4}{3}$,so $f_1 = -\frac{4}{3} f_2$.The equivalent focal length $F$ of the combination is given by $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2}$.Given $F = 60 \, cm$,we substitute the values: $\frac{1}{60} = \frac{1}{-(4/3)f_2} + \frac{1}{f_2} = \frac{1}{f_2} (1 - \frac{3}{4}) = \frac{1}{f_2} (\frac{1}{4})$.Thus,$f_2 = 60 / 4 = 15 \, cm$.Then,$f_1 = -\frac{4}{3} (15) = -20 \, cm$.Therefore,the focal lengths are $-20 \, cm$ and $15 \, cm$.

The ratio of the dispersive powers of two lenses is $4:3$. If the achromatic combination of these two lenses in contact is a convex lens of focal length $60 \, cm$,find the focal lengths of the component lenses.

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