A sphere made of transparent material of refractive index (`mu =3/2` )and of radius 50 cm has a small air bubble 10 cm below the surface. The apparent depth of the bubble if viewed from outside normally is

Options

A. -`50/7` cm
B. `25/7 `cm
C. `100/7` cm
D. `75/7`cm

Answer

To find the apparent depth of the air bubble located 10 cm below the surface of a transparent sphere with a refractive index of \( \mu = \frac{3}{2} \), we can use the formula for apparent depth in a medium. The formula we will use is:

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

Where:
- \( \mu_1 \) is the refractive index of the sphere (which is \( \frac{3}{2} \) or 1.5),
- \( \mu_2 \) is the refractive index of air (which is 1),
- \( u \) is the actual depth of the bubble (which is -10 cm, negative because it is below the surface),
- \( v \) is the apparent depth we want to find,
- \( r \) is the radius of the sphere (which is -50 cm, negative because it is measured from the surface downwards).

### Step 1: Substitute the known values into the formula

Substituting the values into the formula, we have:

\[ \frac{1}{v} - \frac{1.5}{-10} = \frac{1 - 1.5}{-50} \]

### Step 2: Simplify the equation

This simplifies to:

\[ \frac{1}{v} + \frac{1.5}{10} = \frac{-0.5}{-50} \]

### Step 3: Calculate the right-hand side

Calculating the right-hand side:

\[ \frac{-0.5}{-50} = \frac{0.5}{50} = 0.01 \]

### Step 4: Substitute back into the equation

Now we can rewrite the equation:

\[ \frac{1}{v} + 0.15 = 0.01 \]

### Step 5: Isolate \( \frac{1}{v} \)

To isolate \( \frac{1}{v} \):

\[ \frac{1}{v} = 0.01 - 0.15 \]

Calculating this gives:

\[ \frac{1}{v} = -0.14 \]

### Step 6: Solve for \( v \)

Now, we can find \( v \):

\[ v = \frac{1}{-0.14} \approx -7.14 \text{ cm} \]

### Conclusion

Thus, the apparent depth of the bubble when viewed from outside normally is approximately \( 7.14 \) cm below the surface of the sphere.