A thin equiconvex lens has focal length 10 cm and refractive index 1.5 . One of its faces is now silvered and for an object placed at a distance u in front of it, the image coincides with the object. The value of u is

Options

A. 10 cm
B. 5 cm
C. 20 cm
D. 15 cm

Answer

To solve the problem step by step, we will use the lensmaker's formula and the mirror formula. Here's how we can arrive at the solution:

### Step 1: Understanding the Lens
We have a thin equiconvex lens with a focal length \( f = 10 \) cm and a refractive index \( \mu = 1.5 \). Since it is an equiconvex lens, both radii of curvature \( R_1 \) and \( R_2 \) are equal. Let's denote them as \( R \).

### Step 2: Applying the Lensmaker's Formula
The lensmaker's formula is given by:
\[ \frac{1}{f} = \left( \mu - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
For an equiconvex lens, \( R_1 = R \) and \( R_2 = -R \) (the second radius is negative due to the sign convention). Thus, we can write:
\[ \frac{1}{f} = \left( 1.5 - 1 \right) \left( \frac{1}{R} - \frac{-1}{R} \right) = 0.5 \left( \frac{2}{R} \right) \]
This simplifies to:
\[ \frac{1}{f} = \frac{1}{R} \]

### Step 3: Finding the Radius of Curvature
Substituting \( f = 10 \) cm into the equation:
\[ \frac{1}{10} = \frac{1}{R} \]
Thus, we find:
\[ R = 10 \text{ cm} \]

### Step 4: Silvering One Face of the Lens
When one face of the lens is silvered, it behaves like a combination of a lens and a concave mirror. The focal length of the mirror \( f_m \) can be calculated as:
\[ f_m = \frac{R}{2} = \frac{10}{2} = 5 \text{ cm} \]
Since the mirror is silvered on the convex side, its focal length will be negative:
\[ f_m = -5 \text{ cm} \]

### Step 5: Using the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \]
In this case, since the image coincides with the object, we have \( v = -u \). Substituting this into the mirror formula gives:
\[ \frac{1}{-u} + \frac{1}{u} = \frac{1}{-5} \]
This simplifies to:
\[ \frac{-2}{u} = \frac{-1}{5} \]

### Step 6: Solving for Object Distance \( u \)
Cross-multiplying gives:
\[ -2 \cdot 5 = -u \]
Thus:
\[ u = 10 \text{ cm} \]

### Final Answer
The value of \( u \) is \( 10 \) cm.

---