A glass sphere of `(mu=1.5)` with a radius of `15.0 cm` has a tiny air bubble `5 cm` above its centre. The sphere is viewed looking down along the extended radius containing the bubble. What is the apparent depth of the bubble below the surface of the sphere?

Explanation

To find the apparent depth of the bubble below the surface of the glass sphere, we can follow these steps:

### Step 1: Identify the Given Values
- Refractive index of glass, \( \mu_1 = 1.5 \)
- Refractive index of air, \( \mu_2 = 1 \)
- Radius of the sphere, \( R = 15.0 \, \text{cm} \)
- Distance of the bubble above the center, \( OA = 5.0 \, \text{cm} \)

### Step 2: Calculate the Distance from the Center to the Bubble
The distance from the center of the sphere (point O) to the bubble (point A) is given as \( OA = 5.0 \, \text{cm} \). Therefore, the distance from the center to the bottom of the sphere (point B) is:
\[ OB = R - OA = 15.0 \, \text{cm} - 5.0 \, \text{cm} = 10.0 \, \text{cm} \]

### Step 3: Set Up the Sign Convention
According to the sign convention:
- Distances measured upwards (towards the observer) are positive.
- Distances measured downwards (away from the observer) are negative.

Since the bubble is below the surface of the sphere, we take \( U = -OB = -10.0 \, \text{cm} \).

### Step 4: Use the Formula for Apparent Depth
The formula we will use is:
\[ \frac{\mu_2}{V} - \frac{\mu_1}{U} = \frac{\mu_2 - \mu_1}{R} \]
Substituting the known values:
\[ \frac{1}{V} - \frac{1.5}{-10} = \frac{1 - 1.5}{15} \]
This simplifies to:
\[ \frac{1}{V} + \frac{1.5}{10} = \frac{-0.5}{15} \]

### Step 5: Solve for \( \frac{1}{V} \)
Convert the fractions:
\[ \frac{1}{V} + 0.15 = -0.0333 \]
Now, isolate \( \frac{1}{V} \):
\[ \frac{1}{V} = -0.0333 - 0.15 = -0.1833 \]

### Step 6: Calculate \( V \)
Taking the reciprocal gives:
\[ V = \frac{1}{-0.1833} \approx -5.46 \, \text{cm} \]

### Step 7: Interpret the Result
Since the value of \( V \) is negative, it indicates that the apparent depth of the bubble below the surface of the sphere is approximately \( 5.46 \, \text{cm} \).

### Final Answer
The apparent depth of the bubble below the surface of the sphere is approximately \( 8.57 \, \text{cm} \).

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