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in Ray Optics and Optical Instruments by (26.7k points)
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When a point object is placed in front of a spherical refracting surface an image is formed in the refracting medium.

1. 

Complete the ray diagram to locate the position of the image.

2. Obtain the expression \(\frac{n_2}{v}\) - \(\frac{n_1}{u}\) = \(\frac{n_2-n_1}{R}\)

for the position of image inside refracting medium. 

3. If the refracting surface is concave in nature, with the same set up, locate the position of the image by drawing a ray diagram.

1 Answer

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Best answer

1.

2. Refraction at a spherical surface:

Consider a convex surface XY, which separates two media having refractive indices n1 and n2. Let C be the centre of curvature and P be the pole. Let an object is placed at ‘O’, at a distance ‘u’ from the pole.

I is the real image of the object at a distance V from the surface. OA is the incident ray at angle ‘i’ and Al is the refracted ray at an angle ‘r’. OP is the ray incident normally. So it passes without any deviation. From snell’s law,

\(\frac{sin\,i}{sin\,r}=\frac{n_2}{n_1}\)

If ‘i’ and ‘r’ are small, then sin i » i and sin r » r.

i.e., \(\frac{i}{r}=\frac{n_2}{n_1}\)

n1i = n2r ……….(1)

From the ΔOAC, exterior angle = sum of the interior opposite angles

i.e., i = α + θ ………(2)

Similarly, from Δ IAC,

a = r + β

r = α – β ……..(3)

Substituting the values of eq(2) and eq(3)in

eqn.(1) we get,

n1(α + θ) = n2(α – β)
n1α + n1β = n2α – n2β
n1θ + n2β = n2α – n1α
n1θ + n2β = (n2 – n1)α ……(4)

From OAP, we can write,

θ = \(\frac{AP}{OP}\)  [angle = \(\frac{arc}{radius}\)]

From IAP, β = \(\frac{AP}{PI}\) , From CAP , α = \(\frac{AP}{PC}\)

Substituting θ, β and α in equation (4) we get,

\(n_1\frac{AP}{OP}+n_2\frac{AP}{PI}\) = (n2-n1)\(\frac{AP}{PC}\)

\(\frac{n_1}{OP}+\frac{n_2}{PI}\) = \(\frac{(n_2-n_1)}{PC}\)

According to New Cartesian sign convection, we can write,

OP = -u, PI = +v and PC = R

Substituting these values, we get

\(\frac{-n_1}{u}+\frac{n_2}{v}=\frac{n_2-n_1}{R}\)

\(\frac{n_2}{v}-\frac{n_1}{u}\) = \(\frac{n_2-n_1}{R}\)

Case -1:

If the first medium is air, n1 = 1, and n2 = n,

\(\frac{n}{v}-\frac{1}{u}\) = \(\frac{n-1}{R}\)

3.

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