Question

A solid sphere focuses sunlight 5 cm behind itself (measured
from the surface). An object 50 cm in front of the sphere produces an
image 10 cm behind it (both distances measured from the surface).
Determine the radius of the sphere and its refractive index.

Answer 1

Assuming that Sunlight is received in the form of parallel rays. So, object distance = infinity

therefore, focal length of the sphere will be: f = 5 + r

where r is the radius of the sphere.

for the other object,

object distance = u = 50 + r

image distance = v = 10 + r

so, using the lens equation: 1/f = 1/v + 1/u

1/(5 + r) = 1/(10+r) + 1/(50+r)

=> (60 + 2r)(5 + r) = (50 + r)(10 + r)

=> r² + 10r - 200 = 0

solving quadratic equation gives:

r = 10 cm and r = - 20 cm

discarding the negative value gives,

r = 10 cm

this is the radius of the sphere.

the focal length will then be: f = 5 + 10 = 15 cm

using Lens maker's equation:

here, R₁ = r and R₂ = - r

therefore,

=>

=> n = 1.3333.

this is the refractive index of the sphere.