A double convex lens made of glass of refractive index `1.56` has both radii of curvature of magnitude `20 cm`. If an object is placed at a distance of `10 cm` from this lens, find the position of image formed.

Options

A. `22.86` same side of the object
B. `22.86` opposite side of the object
C. `44.89` same side of the object
D. `44.89` opposite side of the object.

Explanation

To find the position of the image formed by a double convex lens, we can follow these steps:

### Step 1: Identify the Given Values
- Refractive index of the lens, \( n = 1.56 \)
- Radii of curvature, \( R_1 = +20 \, \text{cm} \) (since it is convex, we take it as positive)
- \( R_2 = -20 \, \text{cm} \) (the second radius is negative for a double convex lens)
- Object distance, \( u = -10 \, \text{cm} \) (the object distance is negative as per the sign convention)

### Step 2: Use the Lensmaker's Formula
The Lensmaker's formula is given by:

\[ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]

Substituting the values into the formula:

\[ \frac{1}{f} = (1.56 - 1) \left( \frac{1}{20} - \frac{1}{-20} \right) \]

Calculating the right-hand side:

\[ \frac{1}{f} = 0.56 \left( \frac{1}{20} + \frac{1}{20} \right) = 0.56 \left( \frac{2}{20} \right) = 0.56 \left( \frac{1}{10} \right) = \frac{0.56}{10} = 0.056 \]

Thus,

\[ f = \frac{1}{0.056} \approx 17.86 \, \text{cm} \]

### Step 3: Use the Lens Formula to Find Image Distance
The lens formula is given by:

\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]

Rearranging to find \( v \):

\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{u} \]

Substituting the values we found:

\[ \frac{1}{v} = \frac{1}{17.86} + \frac{1}{-10} \]

Calculating each term:

\[ \frac{1}{17.86} \approx 0.056 \quad \text{and} \quad \frac{1}{-10} = -0.1 \]

Now substituting these values:

\[ \frac{1}{v} = 0.056 - 0.1 = -0.044 \]

Thus,

\[ v = \frac{1}{-0.044} \approx -22.72 \, \text{cm} \]

### Step 4: Interpret the Result
The negative sign indicates that the image is formed on the same side as the object. Therefore, the position of the image is approximately \( 22.72 \, \text{cm} \) on the same side as the object.

### Final Answer
The position of the image formed is \( -22.72 \, \text{cm} \) (same side as the object).

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