Equivalent Lens


In this post, we will learn how to find the equivalent focal length or power of the two thin lenses placed in contact with each other.



Suppose we have two thin lenses placed in contacts as shown in the diagram. The object is placed on the right of the combination of lenses at a distance 'u'. The first convex lens forms the image 'I1' of object at a distance of 'v1'. 
Now By Applying Lens formula in 1st lens, we can easily get that


-------------------------(1)

Now the image 'I1' formed by 1st lens acts as a virtual object for 2nd lens and the 2nd lens forms the final image 'I' of the virtual object at a distance of 'v' as can be clearly seen from the main ray diagram.
Now by Applying Lens formula in 2nd lens,  we get:

-------------------------(2)

Now, By adding these two lens equations, we get

-------------------------(3)

If "Feq" is the focal length of the equivalent lens, then

-------------------------(4)

Therefore by eq (3) & (4) we get

This is the required formula for "Focal length" of the Combination of the lenses placed in contacts.

Also, we know that the power of a lens is inversely proportional to its focal length. 
Therefore,

This is the required formula for "Power" of the Combination of the lenses placed in contacts.



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