Name: Reflecting Telescope
Uploaded: 2019-07-19
Duration: 6 min 40 s
Description: This telescope uses a single or a combination of curved mirrors that reflect light and form an image. Watch this amazing video to learn more.

Question

Find an expression for the equivalent focal length for a system of two thin lenses in contact.

Accepted Answer

Assuming two lenses $L_{1}$ and $L_{2}$ with focal length $f_{1}$ and $f_{2}$ are in contact such that their principal axis is same.

Suppose the object is present at point $O$ on the principal axis of the lens. Then its image due to $L_{1}$ will be forming at imaginary $I_{1}$

Thus, $u_{1} = O C_{1}$ and $v_{1} = I_{1} C_{1}$

Then, by lens equation: $- \frac{1}{u_{1}} + \frac{1}{v_{1}} = \frac{1}{f_{1}} \Rightarrow - \frac{1}{O C_{1}} + \frac{1}{I_{1} C_{1}} = \frac{1}{f_{1}}$ ............................ $(1)$

Now when the ray of light passes through $L_{1}$ and enters $L_{2}$ , the object for $L_{2}$ is present at imaginary $I_{1}$ and the image is forming at $I$ .

Thus, $u_{2} = I_{1} C_{1}$ and $v_{2} = I C_{2}$

Then, by lens equation: $- \frac{1}{u_{2}} + \frac{1}{v_{2}} = \frac{1}{f_{2}} \Rightarrow \frac{1}{I C_{2}} - \frac{1}{I_{1} C_{1}} = \frac{1}{f_{2}}$ ............................ $(2)$

Then for the system of the combined lenses we will add $(1)$ and $(2)$

$- \frac{1}{O C_{1}} + \frac{1}{I C_{2}} = \frac{1}{f_{1}} + \frac{1}{f_{2}} = \frac{1}{F}$

$\Rightarrow \frac{1}{F} = \frac{1}{f_{1}} + \frac{1}{f_{2}} \Rightarrow F = \frac{f_{1} f_{2}}{f_{1} + f_{2}}$

Hence, the expression for the equivalent focal length for the given system of lenses will be $F = \frac{f_{1} f_{2}}{f_{1} + f_{2}}$ .

Find an expression for the equivalent focal length for a system of two thin lenses in contact.

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