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 ?

Options

A. `2.5 cm`
B. `-2.5 cm`
C. `5 cm`
D. `-5 cm`

Explanation

To solve the problem of determining the apparent distance of an air bubble in a glass sphere from a convex surface, we will use the formula for refraction through a curved surface. Here’s a step-by-step solution:

### Step 1: Identify the given values
- The distance of the air bubble from the convex surface (object distance, U) = 3 cm (since the bubble is inside the glass, this value is taken as negative in the formula).
- The diameter of the convex surface = 10 cm, thus the radius of curvature (R) = 10 cm / 2 = 5 cm. Since it is a convex surface, R is taken as positive, so R = +5 cm.
- The refractive index of air (n2) = 1 (since the observer is in air).
- The refractive index of glass (n1) = 1.5.

### Step 2: Write the refraction formula
The formula for refraction at a curved surface is given by:
\[ \frac{n_2}{V} - \frac{n_1}{U} = \frac{n_2 - n_1}{R} \]
Where:
- \( V \) = image distance
- \( U \) = object distance (which we will take as negative since the object is inside the medium)
- \( R \) = radius of curvature

### Step 3: Substitute the values into the formula
Substituting the known values into the formula:
\[ \frac{1}{V} - \frac{1.5}{-3} = \frac{1 - 1.5}{5} \]

### Step 4: Simplify the equation
This simplifies to:
\[ \frac{1}{V} + 0.5 = -0.1 \]

### Step 5: Isolate \( \frac{1}{V} \)
Rearranging gives:
\[ \frac{1}{V} = -0.1 - 0.5 \]
\[ \frac{1}{V} = -0.6 \]

### Step 6: Solve for \( V \)
Taking the reciprocal gives:
\[ V = -\frac{1}{0.6} = -\frac{5}{3} \approx -1.67 \text{ cm} \]

### Step 7: Interpret the result
The negative sign indicates that the image of the bubble appears on the same side as the object, which means it appears to be 1.67 cm in front of the convex surface.

### Final Answer
The bubble will appear to be at a distance of approximately 1.67 cm from the convex surface.

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