Wave Optics

JEE Main level, numerical answer, medium

122 of 191

Question 122 of 191

Some distant star is to be observed by some telescope of diameter of objective lens aa, at an angular resolution of 3.0×10−73.0\times 10^{-7} radian. If the wavelength of light from the star reaching the telescope is 500500 nm, the minimum diameter of the objective lens of the telescope is ________ cm. (nearest integer)

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Answer: 203

The angular resolution Δθ\Delta \theta of a telescope is given by the formula:

Δθ=1.22λa\Delta \theta = \dfrac{1.22 \lambda}{a}

where λ\lambda is the wavelength of light and aa is the diameter of the objective lens.

Given:
Δθ=3.0×10−7\Delta \theta = 3.0 \times 10^{-7} rad
λ=500 nm=500×10−9\lambda = 500 \text{ nm} = 500 \times 10^{-9} m =5×10−7= 5 \times 10^{-7} m

Substituting the values into the formula:

3.0×10−7=1.22×5×10−7a3.0 \times 10^{-7} = \dfrac{1.22 \times 5 \times 10^{-7}}{a}

a=1.22×5×10−73.0×10−7a = \dfrac{1.22 \times 5 \times 10^{-7}}{3.0 \times 10^{-7}}

a=6.13.0a = \dfrac{6.1}{3.0} m

a=2.0333...a = 2.0333... m

Converting the diameter into centimeters:

a=2.0333...×100a = 2.0333... \times 100 cm =203.33= 203.33 cm

Rounding to the nearest integer, we get 203203 cm.

Answer: 203203

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