An air bubble in sphere having 4 cm diameter appears 1 cm from surface nearest to eye when looked along diameter If `._(a)mu_(8)=1.5`, the distance of bubble from refracting surface is

Options

A. `1.2 cm`
B. `3.2 cm`
C. `2.8 cm`
D. `1.6 cm`

Answer

To solve the problem step by step, we will use the formula for refraction at a spherical surface:

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

Where:
- \(\mu_2\) = refractive index of the sphere (1.5)
- \(\mu_1\) = refractive index of air (1)
- \(r\) = radius of the sphere (-2 cm, since it is a concave surface)
- \(v\) = image distance (1 cm from the surface nearest to the eye, so -1 cm)
- \(u\) = object distance (distance of the bubble from the refracting surface, which we need to find)

### Step 1: Identify the known values
- \(\mu_2 = 1.5\)
- \(\mu_1 = 1\)
- \(r = -2 \, \text{cm}\) (the radius is negative because it is a concave surface)
- \(v = -1 \, \text{cm}\) (the image distance is negative because the image is on the same side as the object)

### Step 2: Substitute the known values into the refraction formula
\[ \frac{1.5}{-1} - \frac{1}{u} = \frac{1.5 - 1}{-2} \]

### Step 3: Simplify the equation
Calculating the right side:
\[ \frac{1.5 - 1}{-2} = \frac{0.5}{-2} = -0.25 \]

So the equation becomes:
\[ -\frac{1.5}{1} - \frac{1}{u} = -0.25 \]

### Step 4: Rearranging the equation
\[ -1.5 - \frac{1}{u} = -0.25 \]

Adding 1.5 to both sides:
\[ -\frac{1}{u} = -0.25 + 1.5 \]
\[ -\frac{1}{u} = 1.25 \]

### Step 5: Solve for \(u\)
Multiplying both sides by -1:
\[ \frac{1}{u} = -1.25 \]

Taking the reciprocal:
\[ u = -\frac{1}{1.25} = -0.8 \, \text{cm} \]

### Step 6: Final answer
The distance of the bubble from the refracting surface is:
\[ u = -0.8 \, \text{cm} \]