Question

2. (a)
A convex spherical surface of radius +50 mm separates air from glass of refractive index 1.6. A parallel pencil of rays in air is incident on this surface. Find where the refracted pencil comes to a focus in the glass.
(b) Where must a point object be placed in air to make the refracted pencil parallel?
(c) Repeat (a), assuming the incident pencil to be diverging from a point in air 50 cm from the surface.
(d) Repeat (a), assuming the incident pencil to be converging to a point inside the glass 50 mm from the surface.
(e) Repeat (a), assuming the incident pencil to be converging to a point inside the glass 20 mm from the surface (Bennett, chapter 2, question 4).

Can you please explain on sign convention in the answer?

Answer 1

when a parallel beam is incident on a spherical surface. light at the surface is refracted. refractive index of glass n = 1.6 is larger than air n=1.

light at the surface bends towards the normal as shown above. The beam is parallel and every has the same incident angle and refracted at the same angle r. o is the center of the spherical surface. all rays are refracted at the same angle and meet at the point Q

angle PON = i = r +q

from Snells law

sin i/ sin r = n = 1.6 ( refractive index of glass)

for small angle we can approximate Sin i = i and sin r =r

i = nr

from triangle oQN , i = q+r

q = (n-1)r

OQ/Sin(r) = R/Sin(q)

OQ/r = R/ q

OQ = R * r/q = R/(n-1)

The focus point , the point where all the rays meet

= OQ +R = nR/(n-1) = 50*1.6/0.6 = 133.3 mm

b) in case of a point object refraction at spherical surface

refractive index of first surface n₁ =1 ; air

second surface n₂ = 1.6 , glass

image distance  

object distance u

R - radius of surface

refracted beam is parallel , meet at infinity - infinity

-1/u = (1.6 -1) /50  

object distance u = -83.3 mm

-ve sign is , the object is to the left of the surface.

c) incident pencil diverging 50 cm from the surface , i.e the point source object distance u = - 50 cm

put values in the previous formula

1.6/ + 1/50 = 0.6/ 5

= 16 cm , focus

d) converging inside the surface = 20 mm

1.6/20 - 1/u = 0.6/50

u = 14.71 mm , object is to the left of the surface i.e inside the glass surface.

The sign convention in general, distances measured in the direction of light are +ve. distances measured opposite to the direction of light are -ve.

In this case u and v bot are +ve.