A thin convex lens is made of two materials with refractive indices n1...
A thin convex lens is made of two materials with refractive indices n1 and n2, as shown in figure. The radius of curvature of the left and right spherical surfaces are equal. f is the focal length of the lens when n1=n2=n. The focal length is f+Δf when n1=n and n2=n+Δn. Assuming Δn<<(n-1) and 1<n<2, the correct statement(s) is/are:
Given that the lens is made of two plano-convex lenses. Let it be L1 or L2 .
Let focal length of L1=f1,
An focal length of L2=f2
∴ Using lens maker's formula,
1f1=(n1-1)(1R-1∞), and 1f2=(n2-1)(1∞-1(-R))
⇒1f1=n1-1R
⇒1f2=n2-1R
Now, for the first situation, given that,
n1=n2=n and focal length of combination is ‘f’
⇒1f=1f1+1f2=2(n-1)R ....(1)
In the second situation, n1=n, n2=n+Δn and focal length= f+Δf
⇒1f+Δf=n-1R+n+Δn-1R
⇒1f+Δf=2n+Δn-2R .....(2)
Nos dividing equation (1)/(2) we get
f+Δff=2(n-1)2n+Δn-2⇒1+Δff=(2n-2+Δn)-Δn2n+Δn-2
⇒1+Δff=1-Δn2n+Δn-2⇒Δff=-Δn2n+Δn-2
Now, as ′Δn′ is very small ⇒Δff=-Δn2(n-1)      …..(3)
Now, option (A)
n=1.5,  Δn=10-3  and  f=20 cm
⇒|Δf|=f⋅(Δn)2(n-1)=20×10-32×(1.5-1)=0.02 cm⇒(A) is correct.
Option (B)
The relation between Δff or Δnn is dependent of ‘R’
⇒ Remain unchanged for convex or concave surfaces
⇒(B) is correct.
Option (C)
As n<2⇒2(n-1)<n
⇒Δff=12|Δnn-1|>Δnn
⇒Δff>Δnn⇒(C) is incorrect.
Option (D)
If Δnn<0, then Δff>0, using equation (3)
⇒(D) is correct.
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