Question

A ray coming from an object which is situated at zero distance in the air and falls on a spherical glass surface (\( n = 1.5 \)). Then the distance of the image will be ……….  \( R \) is the radius of curvature of a spherical glass.} 

Hint:

For refraction at a spherical surface, use the formula: \[ \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \] For an object at infinity or zero distance, the image distance is determined by the ratio of refractive indices.

Answer 1

Step 1: Using the Refraction Formula for a Spherical Surface 
The refraction formula for a spherical surface is given by: \[ \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \] where:
- \( n_1 \) is the refractive index of the first medium (air, \( n_1 = 1 \)),
- \( n_2 \) is the refractive index of the second medium (glass, \( n_2 = 1.5 \)),
- \( u \) is the object distance,
- \( v \) is the image distance,
- \( R \) is the radius of curvature. 
Step 2: Substituting the Given Values 
Since the object is at \( u = 0 \), the equation simplifies to: \[ \frac{n_2}{v} - \frac{1}{0} = \frac{1.5 - 1}{R} \] Since \( \frac{1}{0} \) tends to infinity, the equation reduces to: \[ \frac{n_2}{v} = \frac{0.5}{R} \] 
Step 3: Solving for \( v \) 
Rearranging the equation: \[ v = \frac{1.5 R}{0.5} = 3R \] Thus, the image distance is \( 3R \).