A convex refracting surface of radius of curvature `20 cm` separates two media of refractive indices `4//3 and 1.60`. An object is placed in the first medium `(mu = 4//3)` at a distance of `200 cm` from the refracting surface. Calculate the position of image formed.

Explanation

To solve the problem of finding the position of the image formed by a convex refracting surface separating two media with different refractive indices, we can use the lens maker's formula for refraction at a spherical surface. Here's a step-by-step solution:

### Step 1: Identify the given values
- Radius of curvature (R) = +20 cm (positive for a convex surface)
- Refractive index of the first medium (μ1) = 4/3 ≈ 1.33
- Refractive index of the second medium (μ2) = 1.60
- Object distance (U) = -200 cm (negative as per the sign convention)

### Step 2: Use the refraction formula
The formula for refraction at a spherical surface is given by:

\[ \frac{\mu_2 - \mu_1}{R} = \frac{\mu_2}{V} - \frac{\mu_1}{U} \]

Where:
- \( V \) = image distance (what we want to find)
- \( U \) = object distance (given as -200 cm)
- \( R \) = radius of curvature (given as +20 cm)

### Step 3: Substitute the values into the formula
Substituting the known values into the formula:

\[ \frac{1.60 - \frac{4}{3}}{20} = \frac{1.60}{V} - \frac{\frac{4}{3}}{-200} \]

### Step 4: Simplify the left-hand side
First, calculate \( \mu_2 - \mu_1 \):

\[ 1.60 - \frac{4}{3} = 1.60 - 1.33 = 0.27 \]

Now, substituting this back into the left-hand side:

\[ \frac{0.27}{20} = \frac{1.60}{V} + \frac{\frac{4}{3}}{200} \]

Calculating \( \frac{0.27}{20} \):

\[ \frac{0.27}{20} = 0.0135 \]

### Step 5: Simplify the right-hand side
Now calculate \( \frac{\frac{4}{3}}{200} \):

\[ \frac{4}{3 \times 200} = \frac{4}{600} = \frac{1}{150} \approx 0.00667 \]

### Step 6: Set up the equation
Now we have:

\[ 0.0135 = \frac{1.60}{V} + 0.00667 \]

### Step 7: Isolate \( \frac{1.60}{V} \)
Subtract \( 0.00667 \) from both sides:

\[ 0.0135 - 0.00667 = \frac{1.60}{V} \]

Calculating the left-hand side:

\[ 0.0135 - 0.00667 = 0.00683 \]

Now we have:

\[ \frac{1.60}{V} = 0.00683 \]

### Step 8: Solve for \( V \)
Now, solve for \( V \):

\[ V = \frac{1.60}{0.00683} \approx 234.15 \text{ cm} \]

### Step 9: Conclusion
The position of the image formed is approximately \( 234.15 \) cm from the refracting surface in the denser medium.