An air bubble in a glass sphere `(mu = 1.5)` is situated at a distance `3 cm` from a convex surface of diameter `10 cm`. At what distance from the surface will the bubble appear ?

Explanation

To solve the problem, we need to determine the apparent position of the air bubble when viewed from outside the glass sphere. We will use the lens maker's formula for refraction at a spherical surface.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Refractive index of glass, \( \mu = 1.5 \)
- Distance of the bubble from the convex surface, \( u = -3 \, \text{cm} \) (the object distance is taken as negative in the sign convention for refraction)
- Radius of curvature of the convex surface, \( R = +5 \, \text{cm} \) (positive because it is a convex surface)

2. **Use the Refraction Formula:**
The formula for refraction at a spherical surface is given by:
\[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \]
Here, \( \mu_1 = 1 \) (refractive index of air), \( \mu_2 = 1.5 \) (refractive index of glass).

3. **Substitute the Values:**
Substituting the values into the formula:
\[ \frac{1.5}{v} - \frac{1}{-3} = \frac{1.5 - 1}{5} \]

4. **Simplify the Equation:**
This simplifies to:
\[ \frac{1.5}{v} + \frac{1}{3} = \frac{0.5}{5} \]
\[ \frac{1.5}{v} + \frac{1}{3} = 0.1 \]

5. **Isolate \( \frac{1.5}{v} \):**
Rearranging gives:
\[ \frac{1.5}{v} = 0.1 - \frac{1}{3} \]
To combine the fractions, convert \( 0.1 \) to a fraction:
\[ 0.1 = \frac{1}{10} \]
Now find a common denominator for \( \frac{1}{10} \) and \( \frac{1}{3} \):
\[ \frac{1}{10} - \frac{1}{3} = \frac{3 - 10}{30} = -\frac{7}{30} \]
Thus,
\[ \frac{1.5}{v} = -\frac{7}{30} \]

6. **Solve for \( v \):**
Rearranging gives:
\[ v = \frac{1.5 \times 30}{-7} = -\frac{45}{7} \approx -6.43 \, \text{cm} \]

7. **Interpret the Result:**
The negative value of \( v \) indicates that the bubble appears to be located at approximately \( 6.43 \, \text{cm} \) on the same side as the observer, which means it appears to be \( 6.43 \, \text{cm} \) away from the surface of the glass.

### Final Answer:
The bubble will appear to be approximately \( 6.43 \, \text{cm} \) from the surface of the glass sphere.