The dispersive powers of the materials of the two lenses are in the ratio 3: 4. If the achromatic combination of these two lenses is of So focal length 60 cm, then the focal lengths of the component lenses are:

Question

The dispersive powers of the materials of the two lenses are in the ratio 3: 4. If the achromatic combination of these two lenses is of So focal length 60 cm, then the focal lengths of the component lenses are: (A) +15 cm and - 20 cm (B) - 15 cm and -20 cm (C) +20 cm and -25 cm (D) -20 cm and +25 cm

Tardigrade Admin · Accepted Answer

Let focal length of the convex and concavelenses be

f_{1}

and

f_{2}

.
In order that this is achromatic combination of lenises, we have

\frac{1}{f _{1}} + \frac{1}{f _{2}} = \frac{1}{F}

Given,

F = 60 c m

∴ \frac{1}{60} = \frac{1}{f _{1}} + \frac{1}{f _{2}} ... (i)

Also, condition for lens combination to be achromatic, we have

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

\Rightarrow \frac{ω _{1}}{ω _{2}} = - \frac{f _{1}}{f _{2}}

where,

ω_{1}

and

ω_{2}

are dispersive powers.
we have given,

\frac{ω _{1}}{ω _{2}} = \frac{3}{4}

∴ \frac{3}{4} = - \frac{f _{1}}{f _{2}}

\Rightarrow f_{2} = - \frac{4}{3} f_{1} ... (ii)

Putting this value in (i), we have

\frac{1}{60} = \frac{1}{f _{1}} - \frac{3}{4 f _{1}}

\Rightarrow f_{1} = + 15 c m

f_{2} = - 20 c m

Hence, one is a convex lens and the other is a concave lens.

Q. The dispersive powers of the materials of the two lenses are in the ratio 3:4. If the achromatic combination of these two lenses is of So focal length 60cm, then the focal lengths of the component lenses are:

+15 cm and - 20 cm

- 15 cm and -20 cm

+20 cm and -25 cm

-20 cm and +25 cm

Q. The dispersive powers of the materials of the two lenses are in the ratio $3 : 4$ . If the achromatic combination of these two lenses is of So focal length $60 c m$ , then the focal lengths of the component lenses are: