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CBSE Class 12 Physics 2016 Outside Delhi Set 1 Paper

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Question : 25
Total: 26
(i) Derive the mathematical relation between refractive indices n1 and n2 of two radii and radius of curvature R for refraction at a convex spherical surface. Consider the object to be a point since lying on the principle axis in rarer medium of refractive index n1 and a real image formed in the denser medium of refractive index n2. Hence, derive lens maker's formula.
(ii) Light from a point source in air falls on a convex spherical glass surface of refractive index 1.5 and radius of curvature 20‌cm. The distance of light source from the glass surface is 100‌cm. At what position is the image formed?
OR
(a) Draw a labelled ray diagram to obtain the real image formed by an astronomical telescope in normal adjustment position. Define its magnifying power.
(b) You are given three lenses of power 0.5 D, 4 D and 10D to design a telescope.
(i) Which lenses should be used as objective and eyepiece? Justify your answer.
(ii) Why is the aperture of the objective preferred to be large? .
Solution:  

(i) Derivation of ‌
n2
v
−‌
n1
u
=‌
(n2−n1)
R

‌
1
f
=(‌
n2−n
n1
)(‌
1
R1
−‌
1
R2
)
Ray diagram showing real image formation as per description
‌θ1=α+β
∴‌‌θ2=β−γ
‌γ=β−θ
For paraxial rays θ1 and θ2 are small
Therefore, n2sin‌θ2=n1sin‌θ2 (Snell's law)
Reduces to
‌
sin‌i
sin‌r
≈‌
i
r
=‌
n2
n1

∴i×n1=r×n2
(α+β)n1=(β−γ)n2
n1(‌
NM
OM
+‌
NM
MC
)‌=(‌
NM
MC
−‌
NM
MI
)n2
n2(‌
1
−u
+‌
1
+R
)‌=(‌
1
+R
−‌
1
u
)n2
‌
n2
v
−‌
n1
u
‌=‌
(n2−n1)
R

Applying above relations to refraction through a lens:

For surface 1
‌
n2−n1
R1
=‌
n2
v′
−‌
n1
u
 ......(i)
For surface 2
‌
n1−n2
R2
=‌
n1
v
−‌
n2
v′
 ......(ii)
Adding eqn. (i) and (ii),
(n2−n1)(‌
1
R1
−‌
1
R2
)=n1(‌
1
v
−‌
1
u
)
For u=∝v=f
∴‌‌
n1
f
‌=(n2−n1)(‌
1
R1
−‌
1
R2
)
⇒‌
1
f
=(‌
n2
n1
−1)(‌
1
R1
−‌
1
R2
)
(iii) R=20‌cm,n2=1.5,n1‌=1,u=−100‌cm
‌
n2
v
‌=‌
(n2−n1)
R
+‌
n1
u

‌=‌
0.5
20‌cm
−‌
1
100‌cm

‌=‌
1.5
100
‌cm
⇒v=100‌cm a real image on the other side, 100‌cm away from the surface.
OR
(a) Labelled ray diagram of Astronomical Telescope
Definition of magnifying power
(b) (i) Identification of lenses Justification
(ii) Reason

Definition-It is the ratio of the angle subtended at the eye, by the final image, to the angle which the object subtends at the lens, or the eye.
(b) (i) Objective =5D
Eye lens =10D
This choice would give higher magnification as
M=‌
f0
fe
=‌
Pe
P0

(ii) High resolving power / Brighter image / lower limit of resolution
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