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Determination of the refractive index of a liquid by pin method using a plane mirror and a convex lens

Theory

Description
 If the object and the image screen be placed on an optical bench that the distance (D) between them is greater than four time the focal length(f) of a given convex lens, then there will be two different positions of the lens for which an equally sharp image will be obtained on the image screen. Let the points O and l and L1 in figure-1 represent respectively the positons of the object and the imge screen and two different positions of the lens for which an equally sharp image is formed. Let distance OI = D and L1 - L2 = x.

From the lens equation, we have
1 ⁄ V - 1 ⁄ U = 1 ⁄ f
1 ⁄ D-U - 1 ⁄ U = 1 ⁄ f [using V + U =0]
Applying sign convention, u is negative
1 ⁄ D-U + 1 ⁄ U = 1 ⁄ f
u2 - ud + df = 0

Solving the above equation which is quadratic, we have two values of corresponding to the two positions of the lens. These are:
u2 = D ⁄ 2 - √(D2 - 4Df ⁄ 2), Position L1 of lens
u2 = D ⁄ 2 + √(D2 - 4Df ⁄ 2), Position L2 of lens
Then x = L1 ~ L2 = u1 ~ u2 = +- √( D2 - 4Df )
or x2 = D2 - 4Df
or f = D2- x2 ⁄ 4D -----(1)
Where D is the distance between the object adn the image msut be greater than 4f and x is the distance between two different positions of lens.
The power of P lens is as usual given by the realtion,
P = 100 ⁄ f (in cm) dioptres

Table-I: Index error (lambda) for D

Length of the index rod in c, (l) Difference of bench scale reading in cm when the two ends of index rod touch the object and screen (d) Index correction for D in cm, lambda = (l-d)
Usually a difference of 0.5

Table-II: Reading for D and x

No of obs. Position of
Objects (O) Image (I) Lens at
L1 L2
Displacement of lens x = L1 ~ L2 Apparent distance between objects and image, D` = O ~ I Corrected distance between object and image, D = D` + lambda
1
2
3

Table-III

No of obs./th> lens displacement (x) from table-II Corrected distance (D) from table-II Focal length,
F =
D2-x2 ⁄ 4D 		Mean focal length (f) (cm) Power, P = 100 ⁄ f dioptres

Discussion

Point Explanation
1 The lens was first used to focus sunlight on paper to get a sharp image; this gave the rough focal length (f).
2 A distance about four times the focal length (4f) was set between the object (illuminated pin) and the screen.
3 The lamp, object pin, lens, and screen were adjusted in a straight line at the same height on the optical bench.
4 The lens was moved to form a clear, inverted image of the pin on the screen — this was the first lens position.
5 Without shifting the object or screen, the lens was moved to the other side to get another sharp image — the second lens position.
6 The positions of the object, lens (both positions), and image were carefully measured and noted.
7 The object was then moved 5 cm backward, and steps 4–6 were repeated for three sets of readings.
8 All readings were entered in Table-II, and the distance (D) and displacement (d) were calculated for each observation.
9 The focal length (f) was found using f = (D² - d²) / 4D, and the power (P) was calculated as P = 100 / f(in cm) diopters.
10 The mean focal length and power were determined from the three observations.