From a point source a light falls on a spherical glass surface (`mu = 1.5` and radius of curvature `= 10 cm`). The distance between point source and glass surface is `50 cm`. The position of image is

Options

A. `25 cm`
B. `50 cm`
C. `100 cm`
D. `150 cm`

Explanation

To find the position of the image formed by a spherical glass surface when light from a point source falls on it, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Refractive index of glass, \( \mu = 1.5 \)
- Radius of curvature, \( R = 10 \, \text{cm} \)
- Distance from the point source to the glass surface, \( u = 50 \, \text{cm} \)

2. **Apply Sign Convention:**
- According to the sign convention, the object distance \( u \) is taken as negative because it is measured in the direction opposite to the incident light. Therefore, \( u = -50 \, \text{cm} \).
- The radius of curvature \( R \) is positive for a convex surface, so \( R = +10 \, \text{cm} \).

3. **Use the Formula for Refraction at a Spherical Surface:**
The formula for refraction at a spherical surface is given by:
\[ -\frac{\mu_1}{u} + \frac{\mu_2}{v} = \frac{\mu_2 - \mu_1}{R} \]
Where:
- \( \mu_1 \) is the refractive index of the first medium (air, \( \mu_1 = 1 \))
- \( \mu_2 \) is the refractive index of the second medium (glass, \( \mu_2 = 1.5 \))
- \( v \) is the image distance we need to find.

4. **Substituting Values into the Formula:**
Substitute the known values into the equation:
\[ -\frac{1}{-50} + \frac{1.5}{v} = \frac{1.5 - 1}{10} \]
Simplifying gives:
\[ \frac{1}{50} + \frac{1.5}{v} = \frac{0.5}{10} \]
This simplifies to:
\[ \frac{1}{50} + \frac{1.5}{v} = 0.05 \]

5. **Rearranging the Equation:**
Rearranging the equation to isolate \( \frac{1.5}{v} \):
\[ \frac{1.5}{v} = 0.05 - \frac{1}{50} \]
Converting \( \frac{1}{50} \) to a decimal gives \( 0.02 \):
\[ \frac{1.5}{v} = 0.05 - 0.02 = 0.03 \]

6. **Solving for \( v \):**
Now, solve for \( v \):
\[ v = \frac{1.5}{0.03} = 50 \, \text{cm} \]

7. **Conclusion:**
The image is formed at a distance of \( 50 \, \text{cm} \) from the spherical surface.

### Final Answer:
The position of the image is \( 50 \, \text{cm} \) from the spherical glass surface.

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