Advertisements
Advertisements
Question
Answer the following question in detail.
What is achromatism? Derive a condition to achieve achromatism for a lens combination. State the conditions for it to be converging.
Solution
- To eliminate chromatic aberrations for extreme colours from a lens, either a convex and a concave lens in contact or two thin convex lenses with proper separation are used.
- This combination is called an achromatic combination. The process of using this combination is termed as achromatism of a lens.
- Let ω1 and ω2 be the dispersive powers of materials of the two-component lenses used in contact for an achromatic combination.
- Let V, R and Y denote the focal lengths for violet, red and yellow colours respectively.
- For lens 1, let
K1 = `(1/"R"_1-1/"R"_2)_1` and K2 = `(1/"R"_1-1/"R"_2)_2` - For the combination to be achromatic, the resultant focal length of the combination must be the same for both the colours,
∴ fV = fR
∴ `1/"f"_"V"=1/"f"_"R"`
For two thin lenses in contact, `1/"f"=1/"f"_1+1/"f"_2`
∴ `1/("f"_"V")_1+1/(("f"_"V")_2)=1/("f"_"R")_1+1/(("f"_"R")_2)` - From Lens maker’s formula,
[(nV)1 − 1]K1 + [(nV)2 − 1]K2 = [(nR)1 − 1]K1 + [(nR)2 −1]K2
∴ `"K"_1/"K"_2=(("n"_"V")_2-("n"_"R")_2)/(("n"_"V")_1-("n"_"R")_1)` ....(1) - Similarly, for mean colour (yellow),
`1/"f"_"Y"=1/(("f"_"Y")_1)+1/(("f"_"Y")_2)` ...(2)
`"K"_1/"K"_2=(("n"_"Y")_2-1)/(("n"_"Y")_1-1)xx(("f"_"Y")_2)/(("f"_"Y")_1)` ....(3) - From equations (1) and (3)
`(("f"_"Y")_2)/(("f"_"Y")_1)=-(("n"_"V")_2-("n"_"R")_2)/(("n"_"V")_1-("n"_"R")_1)xx(("n"_"Y")_2-1)/(("n"_"Y")_1-1)` ...(4)
Now, dispersive power ω1 = `(("n"_"V")_2-("n"_"R")_2)/(("n"_"Y")_2-1)` and
ω2 = `(("n"_"V")_2-("n"_"R")_2)/(("n"_"Y")_2-1)` - Substituting values of ω1 and ω2 in equation (4), we get,
`(("f"_"Y")_2)/(("f"_"Y")_1)=ω_2/ω_1`
This is the condition for achromatism of a combination of lenses.
Condition for converging:
For this combination to be converging, fY must be positive.
Using equation (3), for fY to be positive,
(fY)1 < (fY)2 ⇒ ω1 < ω2
APPEARS IN
RELATED QUESTIONS
Answer the following question.
Explain spherical aberration for spherical mirrors. How can it be minimized? Can it be eliminated by some curved mirrors?
Explain chromatic aberration for spherical lenses.
Answer the following question in detail.
Describe spherical aberration for spherical lenses. What are the different ways to minimize or eliminate it?
Answer the following question in detail.
Derive the expressions for the magnifying power and the length of a compound microscope using two convex lenses.
Refractive index of a flint glass varies from 1.60 to 1.66 for visible range. Radii of curvature of a thin convex lens are 10 cm and 15 cm. Calculate the chromatic aberration between extreme colours.
Refractive index of a flint glass varies from 1.60 to 1.66 for visible range. Radii of curvature of a thin convex lens are 10 cm and 12 cm. The chromatic aberration between extreme colours is ____________.
A convex lens of focal length 'f' is placed in contact with a concave lens of the same focal length. The equivalent focal length of the combination is ______.
Focal length of a convex lens will be minimum for ______.
The equiconvex lens has a focal length 'f'. If the lens is cut along the line perpendicular to principal axis and passing through the pole, what will be the focal length of any half part?
A glass slab of thickness 4 cm contains the same number of waves as in 'x' cm of water column when both are traversed by the same monochromatic light. If the refractive indices of glass and water for that light are `5/3` and `4/3` respectively, the value of x will be ______.
