An air bubble is seen inside a solid sphere of glass `(n=1.5)` of `4.0 cm` diameter at a distance of `1.0 cm` from the surface of the sphere (on seeing along the diameter). Determine the real position of the bubble inside the sphere.

Explanation

To determine the real position of the air bubble inside the glass sphere, we can use the lens maker's formula for refraction. Here's the step-by-step solution:

### Step 1: Understand the given data
- The refractive index of glass, \( n_1 = 1.5 \)
- The refractive index of air (the bubble), \( n_2 = 1.0 \)
- Diameter of the glass sphere = 4.0 cm, hence the radius \( r = \frac{4.0}{2} = 2.0 \) cm
- Distance of the bubble from the surface of the sphere = 1.0 cm

### Step 2: Calculate the distance of the bubble from the center of the sphere
Since the bubble is 1.0 cm from the surface of the sphere, the distance from the center of the sphere to the bubble is:
\[ \text{Distance from center to bubble} = r - \text{distance from surface} = 2.0 \, \text{cm} - 1.0 \, \text{cm} = 1.0 \, \text{cm} \]

### Step 3: Set up the sign convention
- The direction towards the observer (along the diameter) is considered positive.
- The distance of the bubble from the center of the sphere is positive (1.0 cm).
- The radius of the sphere is negative when considering the direction of the light rays going out of the sphere.

### Step 4: Use the lens maker's formula
Using the formula for refraction at a spherical surface:
\[ \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{r} \]
where:
- \( n_1 = 1.5 \) (glass)
- \( n_2 = 1.0 \) (air)
- \( u \) is the object distance (distance of the bubble from the center)
- \( v \) is the image distance (distance from the center to the observer)
- \( r = -2.0 \) cm (radius of the sphere)

### Step 5: Substitute the known values into the formula
Let \( u = 1.0 \, \text{cm} \) (distance from the center to the bubble):
\[ \frac{1.0}{v} - \frac{1.5}{1.0} = \frac{1.0 - 1.5}{-2.0} \]
This simplifies to:
\[ \frac{1.0}{v} - 1.5 = \frac{-0.5}{-2.0} \]
\[ \frac{1.0}{v} - 1.5 = 0.25 \]

### Step 6: Solve for \( v \)
Rearranging gives:
\[ \frac{1.0}{v} = 1.5 + 0.25 = 1.75 \]
Thus,
\[ v = \frac{1.0}{1.75} \approx 0.571 \, \text{cm} \]

### Step 7: Interpret the result
The value of \( v \) indicates the position of the bubble as seen by the observer. Since \( v \) is positive, it means the bubble is located approximately 0.571 cm from the center of the sphere towards the observer.

### Final Answer
The real position of the bubble inside the sphere is approximately **0.571 cm** from the center of the sphere.