Name: Total Internal Reflection From Prism
Uploaded: 2019-07-19
Duration: 1 min 56 s
Description: You must have seen reflectors on a bicycle. Why are they attached to the mudguards? What use do they have? Let’s watch this video on reflectors to find out.

Question

Derive the expression for combined focal length of two thin lenses kept in contact.

Accepted Answer

Let two thin lenses $L_{1}$ and $L_{2}$ of focal lengths $f_{1}$ and $f_{2}$ be put in contact. O is a point object at a distance u from the lens $L_{1}$ Its image is formed at I after refraction through the two lenses at a distance v from the combination. The lens L1 forms the image of O at $I^{'}$ , $I^{'}$ then serves as a virtual object for the lens $L_{2}$ which forms a real image at I.

Now, we know that by lens formula

$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$

Applying this relation for refraction at the lens $L_{1}$ of focal length $f_{1}$ can written as

$\frac{1}{v^{'}} - \frac{1}{u} = \frac{1}{f_{1}} . . . . . . (1)$

For refraction at the lens $L_{2}$

$u = v^{'}, v = v$

For this lens

$\frac{1}{v} - \frac{1}{v^{'}} = \frac{1}{f_{2}} . . . . (2)$

And if is the focal length of the combination, then

$\frac{1}{v} - \frac{1}{u} = \frac{1}{F} . . . . . . (3)$

Adding (1) and (2), we

$\frac{1}{v^{'}} - \frac{1}{u} + \frac{1}{v} - \frac{1}{v^{'}} = \frac{1}{f_{1}} + \frac{1}{f_{2}} \frac{1}{F} = \frac{1}{f_{1}} + \frac{1}{f_{2}}$

Derive the expression for combined focal length of two thin lenses kept in contact.

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