An achromatic convergent doublet of two lenses in contact has a power +2 D . The convex lens has power +5 D . The ratio of the dispersive powers of the convergent and divergent lenses is (in magnitude

Question

An achromatic convergent doublet of two lenses in contact has a power +2 D . The convex lens has power +5 D . The ratio of the dispersive powers of the convergent and divergent lenses is (in magnitude)

TestHub · Accepted Answer

The power of the achromatic doublet is given by $P = P_{1} + P_{2}$ .

Given $P = + 2 D$ and the power of the convex (convergent) lens $P_{1} = + 5 D$ .

Substituting the values:

$+ 2 = + 5 + P_{2}$

$P_{2} = - 3 D$

For an achromatic doublet, the condition for no chromatic aberration is:

$\frac{ω _{1}}{f _{1}} + \frac{ω _{2}}{f _{2}} = 0$

Since power $P = \frac{1}{f}$ , this can be written as:

$ω_{1} P_{1} + ω_{2} P_{2} = 0$

$ω_{1} (5) + ω_{2} (- 3) = 0$

$5 ω_{1} = 3 ω_{2}$

$\frac{ω _{1}}{ω _{2}} = \frac{3}{5}$

The ratio of the dispersive powers of the convergent and divergent lenses is $3 : 5$ .

Answer: $3 : 5$

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