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 solid glass sphere, we can use the formula for refraction through a spherical surface. Here are the steps to solve the problem:

### Step 1: Identify the Given Data
- Diameter of the glass sphere = 4.0 cm
- Radius of the sphere (R) = Diameter / 2 = 4.0 cm / 2 = 2.0 cm
- Distance of the bubble from the surface of the sphere = 1.0 cm
- Refractive index of glass (n) = 1.5
- Refractive index of air (n') = 1.0

### Step 2: Determine the Image Distance (v)
Since the bubble is seen at a distance of 1.0 cm from the surface of the sphere, we need to find the image distance (v) from the center of the sphere. The distance from the center of the sphere to the surface is the radius (2.0 cm), so:
- Image distance, v = Radius - Distance from surface = 2.0 cm - 1.0 cm = 1.0 cm
- Since the image is formed on the same side as the object, we take v as negative: v = -1.0 cm.

### Step 3: Calculate the Object Distance (u)
Using the refraction formula for a spherical surface:
\[ \frac{n'}{v} - \frac{n}{u} = \frac{n - n'}{R} \]
Where:
- n' = refractive index of air = 1.0
- n = refractive index of glass = 1.5
- R = radius of the sphere = 2.0 cm (but we take it as negative since the center of curvature is on the opposite side of the incident light)

Substituting the known values:
\[ \frac{1.0}{-1} - \frac{1.5}{u} = \frac{1.5 - 1.0}{-2} \]

### Step 4: Simplify the Equation
This simplifies to:
\[ -1 - \frac{1.5}{u} = \frac{0.5}{-2} \]
\[ -1 - \frac{1.5}{u} = -0.25 \]

### Step 5: Rearranging the Equation
Rearranging gives:
\[ -\frac{1.5}{u} = -0.25 + 1 \]
\[ -\frac{1.5}{u} = 0.75 \]

### Step 6: Solve for u
Multiplying both sides by -u:
\[ 1.5 = -0.75u \]
\[ u = -\frac{1.5}{0.75} = -2.0 \text{ cm} \]

### Step 7: Determine the Real Position of the Bubble
The negative sign indicates that the object (the bubble) is located inside the sphere. The real position of the bubble from the center of the sphere is 2.0 cm.

### Final Answer
The real position of the bubble inside the sphere is 2.0 cm from the center of the sphere.

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