Question

Two light bulbs are $1.0m$apart. from, what distance can these light bulbs be marginally resolved by a small telescope with a $4.0-cm-diameter$objective lens? assume that the lens is diffraction limited and$\lambda =600nm$.

Short answer

Light bulbs be marginally resolved by a small telescope with a $4.0-cm-diameter$objective lens with distance$L=55.5Km$.

Step-by-step solution

Given information 

We have given that:
Distance between two bulbs is$d=1.0cm$,

the diameter islocalid="1648894941098" $D=4.0cm$,

the wavelength islocalid="1648894948467" $\lambda =600nm\to 600\times {10}^{-9}m$.

We need to find the distance.

Simplification

Releigh's criteria for angular resolution for circular diameter, $D>>h,is$:

$d\theta =\frac{1.22\lambda }{D}$

$d\theta =\frac{1.22\times 600\times {10}^{-9}}{0.04}$

$d\theta =18\times {10}^{-6}$

Now we have to find L from,
$d\theta =\frac{d}{L}$

By multiplying both the sides by L,

localid="1648895025315" $L\times d\theta =\cancel{L}$localid="1648895031802" $\times \frac{d}{\cancel{L}}$

localid="1648895036807" $L\times d\theta =d$.

Dividing both the sides by $d\theta$,

localid="1648895043506" $\frac{L\times \cancel{d\theta }}{\cancel{d\theta }}=\frac{d}{d\theta }$

$L=\frac{d}{d\theta }$.

Substitute the values in equation,

$L=\frac{1.0}{18x{10}^{-6}}$

$L=55.5km.$

Here, $L$is the distance that these light bulbs can be marginally resolved by a small telescope.