Question

The aperture diameter of a telescope is $5\text{m}$. The separation between the moon and the earth is $4\times {10}^5\text{km}$. With light of wavelength of $5500\mathring{\text{A}}$, the minimum separation between objects on the surface of the moon, so that they are just resolved, is close to:

• A. $60\text{m}$
• B. $20\text{m}$
• C. $200\text{m}$
• D. $600\text{m}$

Answer 1

The correct option is A $60\text{m}$


Smallest angular separation between two distant point objects, is,

$d\theta =1.22\frac{\lambda }{a}...\left(1\right)$

$a$ = aperture diameter of telescope.

Now, $\text{angle in radians}=\frac{\text{arc length}}{\text{radius}}$

As, the angle $d\theta$ is very small, we can write,

Distance $d\theta =\frac{O_1{O}_2}{d}...\left(2\right)$

Equating $\left(1\right)$ and $\left(2\right)$, we get,

$1.22\frac{\lambda }{a}=\frac{O_1{O}_2}{d}$

$\therefore O_1{O}_2=\left(1.22\frac{\lambda }{a}\right)d$

$=\frac{\left(1.22\right)(5500\times {10}^{-10})\times 4\times {10}^5\times {10}^3}{5}$

$=5368\times {10}^{-2}\text{m}=53.68\text{m}\approx 60\text{m}$

Hence, $\left(A\right)$ is the correct answer.