Achromatic combination of lenses comprises two ...

Hint: This problem can be solved by the direct formula for the distance between two lenses in an achromatic combination in terms of the focal lengths of the two lenses. The distance between the two lenses should be equal to half of the sum of the focal lengths of the two lenses.

Formula used: $d=\dfrac{{{f}_{1}}+{{f}_{2}}}{2}$

Complete step by step answer:
We will solve this problem by using the direct formula for the distance between two lenses made of the same material in an achromatic combination. The distance must be equal to half of the sum of the focal lengths of the two lenses. Therefore, let's express this formula clearly.
For an achromatic combination of two lenses of the same material, the distance $d$ between the two lenses is given by
$d=\dfrac{{{f}_{1}}+{{f}_{2}}}{2}$ --(1)
Where ${{f}_{1}},{{f}_{2}}$ are the focal lengths of the two lenses respectively.
Now, let us analyze the question.
The distance between the two lenses in the achromatic combination is given to be $d=4cm$.
The focal length of one of the lenses is ${{f}_{1}}=5cm$.
Let the focal length of the other lens be ${{f}_{2}}$.
Therefore, using (1), we can derive the focal length of the other lens, that is, ${{f}_{2}}$ as follows:
$4=\dfrac{5+{{f}_{2}}}{2}$
$\therefore 4\times 2=5+{{f}_{2}}$
$\therefore 8=5+{{f}_{2}}$
$\therefore {{f}_{2}}=8-5=3cm$
Thus, we have determined the focal length of the second lens as $3cm$.

So, the correct answer is 'Option D'.

Note: It is essential for students to understand that an achromatic combination of two lenses, also known as an achromatic doublet, is composed of a convex lens and a concave lens. Therefore, the focal length of the convex lens is positive while that of the concave lens is negative according to the sign convention. However, while applying formula (1) for calculations, one must take the magnitudes of these focal lengths without considering their signs; otherwise, incorrect results may arise. It may even result in a scenario where the distance between the lenses is calculated to be negative, which is physically impossible.

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Two thin prisms are combined to form an achromatic ...

Achromatic prisms are formed by combining two prisms with varying refractive indices to minimize chromatic aberrations. For example, consider two prisms,

P₁(μ_v=1.523, μ_r=1.515) and P₂(μ_v=1.666, μ_r=1.650),

this combination can be engineered to achieve a balance in dispersion.

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