A symmetric biconvex lens of radius of curvature R and made of glass of refractive index 1.5 is placed on a layer of liquid placed on top of a plane mirror as shown in the figure. In optical needle with it tip on the principal axis of the lens is moved along the axis until its real ,inverted, and image coincides with needle itself.The distance of the needle from lens is measured to be x . On removing the liquid layer and repeating the experiment ,the distance is found to be y.Obtain the expression for the refractive index of the liquid in terms of x and y
Answers
The liquid acts as a mirror. Focal length of the liquid is 2.
Focal length of the system (convex lens + liquid), = 45 .
For a pair of optical systems placed in contact, the equivalent focal length is given as:
1
=
1
1
+
1
2
→ 2 =
∙ 1
1 −
=
45 ∙ 30
30 − 45 = −90 .
Let the refractive index of the lens be 1and the radius of curvature of one surface be . Hence, the radius
of curvature of the other surface is −. can be obtained using the relation:
1
1
= (1 − 1) (
1
−
1
(−)
) → = 21
(1 − 1) = 2 ∙ 30 ∙ (1.5 − 1) = 30 .
Let 2 be the refractive index of the liquid.
Radius of curvature of the liquid on the side of the plane mirror is ∞ .
Radius of curvature of the liquid on the side of the lens, − = −30 .
The value of 2 can be calculated using the relation:
1
2
= (2 − 1) (
1
(−)
−
1
∞
) → 2 = 1 −
2
= 1 −
30
(−90)
=
4
3
≈ 1.33.
Hey !!
When a liquid is placed on top of the plane mirror and convex lens over it, then this whole system would become a combination of convex lens of glass and planoconcave lens of liquid. This is shown in the figure.
Let focal length of convex lens = f₁
Focal length of planoconcave liquid lens = f₂
combined focal length = F
In both cases image coincides with needle, hence ray is normal to plane mirror. So, needle position is focal lengths of convex lens and combined system respectively.
Now, According to the question,
f₁ = y unit
F = x unit
We also know that for combination of two lenses
1/F = 1/f₁ + 1/f₂
= 1/f₁ = 1/F - 1/f₁
= 1/f₂ = 1/x - 1/y
f₂ = xy / y - x
For glass lens, let R₁ = R, R₂ = -R. From lens maker formula
1/f = (n-1) (1/R - (1/R))
= 1/y = (1.5 - 1) (1/R + 1/R)
= 1/y = 1/R
R = y unit
For liquid planoconcave lens
R₁ = -R , R₂ = infinity
For lens maker formula
1/f₂ = (n₁ - 1) ( -1/R - 1/infinity)
= 1 - n₁ = y - x / x
n₁ = 1 - y - x / x
n₁ = x - y + z / x
n₁ = 2x - y / x
GOOD LUCK !!
