Question

Light from a point source in air falls on a spherical glass surface (refractive index, \( \mu = 1.5 \) and radius of curvature \( R = 50 \) cm). The image is formed at a distance of 200 cm from the glass surface inside the glass. The magnitude of distance of the light source from the glass surface is 1cm.

Hint:

Apply the formula for refraction at a spherical surface, \( \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \). Be careful with the sign conventions for \( u \), \( v \), and \( R \). For a convex surface, \( R \) is positive when light goes from a rarer to a denser medium. Real objects have negative \( u \), and real images formed on the opposite side of the refracting surface have positive \( v \). Ensure consistent units throughout the calculation and convert the final answer to the required unit.

Answer 1

Correct Answer: 4

We will use the formula for refraction at a spherical surface: \[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \] where: \( \mu_1 \) is the refractive index of the first medium (air) = 1 
\( \mu_2 \) is the refractive index of the second medium (glass) = 1.5 
\( u \) is the object distance from the spherical surface (to be found) 
\( v \) is the image distance from the spherical surface = 200 cm (positive as the image is formed in the second medium) 
\( R \) is the radius of curvature of the spherical surface = +50 cm (positive as the surface is convex to the incident light) 
Substituting the given values into the formula: \[ \frac{1.5}{200} - \frac{1}{u} = \frac{1.5 - 1}{50} \] \[ \frac{1.5}{200} - \frac{1}{u} = \frac{0.5}{50} \] \[ \frac{1.5}{200} - \frac{1}{u} = \frac{1}{100} \] \[ \frac{3}{400} - \frac{1}{u} = \frac{1}{100} \] \[ \frac{1}{u} = \frac{3}{400} - \frac{1}{100} = \frac{3}{400} - \frac{4}{400} = -\frac{1}{400} \] \[ u = -400 \, \text{cm} \] The negative sign indicates that the object is real and located on the side from which the light is incident. 
The magnitude of the distance of the light source from the glass surface is \( |u| = 400 \, \text{cm} \). 
To convert this distance to meters, we divide by 100: \[ |u| = \frac{400}{100} \, \text{m} = 4 \, \text{m} \]