A glass sphere of radius 5 cm has a small bubble at a distance 2 cm from its centre.The bubble is viewed along a Diameter of the sphere from the side on which it lies.How far from the surface will it appear.Refractive index of glass is 1.5
2.5 cm behind the surface

Options

A. 2.5 cm
B. 5.2 cm
C. `-5.2` cm
D. `-2.5` cm

Answer

To solve the problem step by step, we need to apply the lens maker's formula for refraction at a curved surface. Here’s how we can approach the solution:

### Step 1: Understand the setup
We have a glass sphere with a radius of 5 cm and a bubble located 2 cm from the center of the sphere. The refractive index of glass is given as 1.5. The bubble is viewed from the side of the sphere where it is located.

### Step 2: Identify the distances
- The radius of the sphere (R) = 5 cm
- The distance of the bubble from the center of the sphere (u) = 2 cm
- The distance from the surface of the sphere to the center = radius - distance of the bubble from the center = 5 cm - 2 cm = 3 cm.

### Step 3: Use the lens maker's formula
The formula for refraction at a curved surface is given by:

\[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \]

Where:
- \( \mu_1 \) = refractive index of the medium from which the light is coming (air, \( \mu_1 = 1 \))
- \( \mu_2 \) = refractive index of the medium into which the light is entering (glass, \( \mu_2 = 1.5 \))
- \( u \) = object distance (distance of the bubble from the center of the sphere, which is -2 cm since it is on the same side as the incoming light)
- \( v \) = image distance (what we need to find)
- \( R \) = radius of curvature (positive as the center of curvature is on the same side as the incoming light, \( R = 5 \) cm)

### Step 4: Substitute the values into the formula
Substituting the known values into the formula:

\[ \frac{1.5}{v} - \frac{1}{-2} = \frac{1.5 - 1}{5} \]

This simplifies to:

\[ \frac{1.5}{v} + \frac{1}{2} = \frac{0.5}{5} \]

### Step 5: Simplify and solve for \( v \)
Now we can simplify the equation:

\[ \frac{1.5}{v} + \frac{1}{2} = 0.1 \]

To isolate \( \frac{1.5}{v} \):

\[ \frac{1.5}{v} = 0.1 - 0.5 \]
\[ \frac{1.5}{v} = -0.4 \]

Now, cross-multiply to solve for \( v \):

\[ 1.5 = -0.4v \]
\[ v = \frac{1.5}{-0.4} = -3.75 \text{ cm} \]

### Step 6: Interpret the result
The negative sign indicates that the image is formed on the same side as the object (the bubble). The distance from the surface of the sphere to the image is:

\[ \text{Distance from surface} = \text{Radius of sphere} - |v| = 5 \text{ cm} - 3.75 \text{ cm} = 1.25 \text{ cm} \]

### Final Answer
The bubble will appear 1.25 cm from the surface of the glass sphere.

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