Question 176: Wave Optics

Question

Some distant star is to be observed by some telescope of diameter of objective lens $a$, at an angular resolution of $3.0 	imes 10^{-7}$ radian. If the wavelength of light from the star reaching the telescope is 500 nm , the minimum diameter of the objective lens of the telescope is $\_\_\_\_$ cm. (nearest interger)

Accepted Answer

According to Rayleigh criterion the angular resolution ( $	heta$ ) of a telescope is the smallest angle between two distant objects that can be seen as separate points. For a circular aperture, the formula is:
$$ 	heta=\frac{1.22 \lambda}{a} $$
Where $	heta=$ Angular resolution in radians, $\lambda=$ Wavelength of the light and $\mathrm{a}=$ Diameter of the objective lens.
We need to find the diameter of the objective lens,
$$ a=\frac{1.22 \lambda}{	heta} $$
The wavelength of light is $\lambda=500 \mathrm{~nm}=500 	imes 10^{-9} \mathrm{~m}$.
The angular resolution of telescope is $	heta=3.0 	imes 10^{-7} \mathrm{rad}$.
Substituting the values,
$$ \mathrm{a}=\frac{1.22 	imes\left(500 	imes 10^{-9}ight)}{3.0 	imes 10^{-7}} $$
$$\Rightarrow $$ $\mathrm{a}=\frac{610 	imes 10^{-9}}{3.0 	imes 10^{-7}}$
$$\Rightarrow $$ $a=\frac{610}{3} 	imes 10^{-2} \mathrm{~m}$
$$\Rightarrow $$ $a \approx 203.33 	imes 10^{-2} \mathrm{~m}$
$$\Rightarrow $$ $\mathrm{a}=2.0333 \mathrm{~m} 	imes 100$
$$\Rightarrow $$ $$ \mathrm{a}=203.33 \mathrm{~cm} $$
Therefore, the minimum diameter of the objective lens is 203 cm .

Wave Optics

Integer / Numeric Answer Type