Describe the astronomical telescope on the basis of the following points:
(i) Labelled ray diagram.
(ii) Derivation of formula for magnifying power, when final image is formed at the least distance of distinct vision.
Where, O→ Objective lens, O′→ Eye lens, PQ→ Image
(ii) Expression for magnifying power: If angle made by an object at least distance of distinct vision is α and angle made by an image is β. Then magnification.
m=βα .........(i)
Object is situated at infinity, therefore angle subtended by an object on objective lens is said to be angle subtended by an object on eye.
Now, tanβ=P′Q′O′Q′ and tanα=P′Q′OQ′
As α and β are very small hence,
tanβ≈β and tanα≈α
∴β=P′Q′O′Q′ and α=P′Q′OQ′
Putting the values of β and α in equation (i), we get
m=P′Q′/O′Q′P′Q′/OQ′=OQ′O′Q′ .........(ii)
By sign convention distance of P′Q′ from Le=uc(−vc)=O′Q′ and focal length of objective lens Lo is
f0=OQ′ (+ve)
∴m=f0−uc ........(iii)
When image is formed at least distance of distinct vision
From formula: 1f=1v−1u
or 1fe=1−ve−1−ue
or 1fe=−1ve+1ue
But, ve=D
∴1fe=−1D+1ue
or 1ue=1fe+1D
Substituting above value in equation (iii),
m=−f0(1fe+1D)
m=−f0fe(1+feD)