A glass sphere of radius 15 cm has a small bubble 6 cm from its centre. The bubble is viewed along a diameter of the sphere from the side on which it lies. How for from the surface will it appear to be if the refractive index of glass is 1.5 ?

Answer

To solve the problem step by step, we will use the concepts of optics, specifically the refraction of light through a spherical medium.

### Step 1: Understand the Given Information
We have a glass sphere with:
- Radius (R) = 15 cm
- A small bubble located 6 cm from the center of the sphere.

### Step 2: Determine the Object Distance (u)
The object distance (u) is measured from the center of the sphere to the bubble. Since the bubble is 6 cm from the center, we can find the distance from the surface of the sphere to the bubble:
- Distance from the center to the surface = Radius = 15 cm
- Therefore, the distance from the surface to the bubble (u) = 15 cm - 6 cm = 9 cm.

Since we are using the sign convention where distances measured in the direction of incident light are negative, we have:
- u = -9 cm.

### Step 3: Identify the Refractive Indices
- Refractive index of air (μ1) = 1.0
- Refractive index of glass (μ2) = 1.5

### Step 4: Apply the Refraction Formula
We will use the formula for refraction at a spherical surface:
\[ \frac{\mu_1}{v} - \frac{\mu_2}{u} = \frac{\mu_1 - \mu_2}{R} \]

Substituting the known values:
\[ \frac{1}{v} - \frac{1.5}{-9} = \frac{1 - 1.5}{15} \]

### Step 5: Simplify the Equation
This simplifies to:
\[ \frac{1}{v} + \frac{1.5}{9} = -\frac{0.5}{15} \]

Calculating the right side:
\[ -\frac{0.5}{15} = -\frac{1}{30} \]

Now, we have:
\[ \frac{1}{v} + \frac{1.5}{9} = -\frac{1}{30} \]

### Step 6: Calculate \(\frac{1.5}{9}\)
Calculating \(\frac{1.5}{9}\):
\[ \frac{1.5}{9} = \frac{1}{6} \]

### Step 7: Substitute and Solve for \(\frac{1}{v}\)
Now substituting back:
\[ \frac{1}{v} + \frac{1}{6} = -\frac{1}{30} \]

To solve for \(\frac{1}{v}\), we need a common denominator. The common denominator for 6 and 30 is 30:
\[ \frac{1}{v} + \frac{5}{30} = -\frac{1}{30} \]

Subtracting \(\frac{5}{30}\) from both sides:
\[ \frac{1}{v} = -\frac{1}{30} - \frac{5}{30} = -\frac{6}{30} = -\frac{1}{5} \]

### Step 8: Find v
Taking the reciprocal gives:
\[ v = -5 \text{ cm} \]

### Step 9: Interpret the Result
The negative sign indicates that the image is virtual and located on the same side as the object (the bubble).

### Step 10: Calculate the Distance from the Surface
Since the radius of the sphere is 15 cm, the distance of the virtual image from the surface of the sphere is:
\[ \text{Distance from surface} = 15 cm - 5 cm = 10 cm. \]

### Final Answer
The bubble will appear to be 10 cm from the surface of the glass sphere.

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