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.

Options

A. `120 cm`
B. `240 cm`
C. `100 cm`
D. `60 cm`

Explanation

To solve the problem of finding the position of the image formed by a convex refracting surface, we can follow these steps:

### Step 1: Identify the Given Values
- Refractive index of the first medium (n1) = 4/3
- Refractive index of the second medium (n2) = 1.60
- Radius of curvature (R) = 20 cm (for a convex surface, R is positive)
- Object distance (u) = -200 cm (the object distance is negative as per the sign convention)

### Step 2: Use the Refraction Formula
For a spherical refracting surface, the formula is given by:
\[ -\frac{n_1}{u} + \frac{n_2}{v} = \frac{n_2 - n_1}{R} \]

### Step 3: Substitute the Known Values into the Formula
Substituting the values we have:
\[ -\frac{4/3}{-200} + \frac{1.6}{v} = \frac{1.6 - 4/3}{20} \]

### Step 4: Simplify the Left Side
Calculating the left side:
\[ \frac{4/3}{200} = \frac{4}{600} = \frac{1}{150} \]
So, we have:
\[ \frac{1}{150} + \frac{1.6}{v} = \frac{1.6 - 1.3333}{20} \]
Calculating \(1.6 - 1.3333\):
\[ 1.6 - 1.3333 = 0.2667 \]
Now substituting this into the equation:
\[ \frac{1}{150} + \frac{1.6}{v} = \frac{0.2667}{20} \]

### Step 5: Calculate the Right Side
Calculating the right side:
\[ \frac{0.2667}{20} = 0.013335 \]

### Step 6: Rearranging the Equation
Now we have:
\[ \frac{1.6}{v} = 0.013335 - \frac{1}{150} \]
Calculating \(\frac{1}{150}\):
\[ \frac{1}{150} = 0.0066667 \]
So, substituting this value:
\[ \frac{1.6}{v} = 0.013335 - 0.0066667 = 0.0066683 \]

### Step 7: Solve for v
Now, rearranging gives:
\[ v = \frac{1.6}{0.0066683} \approx 240.00 \text{ cm} \]

### Conclusion
The position of the image formed is approximately **240 cm** from the refracting surface.

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