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

Explanation

To solve the problem step by step, we will follow the concepts of optics related to lenses and mirrors.

### Step 1: Understand the problem
We have a thin equiconvex lens with a focal length (f) of 10 cm and a refractive index (n) of 1.5. One of its faces is silvered, effectively turning it into a lens-mirror combination. We need to find the distance (u) from the lens to the object, given that the image coincides with the object.

### Step 2: Determine the radius of curvature (R)
For an equiconvex lens, both radii of curvature are equal (R1 = R2 = R). The lens formula for a thin lens is given by:

\[ \frac{1}{f} = \left( \frac{n}{n_0} - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]

Where:
- \( n_0 \) is the refractive index of air (approximately 1).
- \( n \) is the refractive index of the lens (1.5).

Since R1 is positive and R2 is negative for a convex lens, we can write:

\[ \frac{1}{f} = \left( \frac{1.5}{1} - 1 \right) \left( \frac{1}{R} + \frac{1}{R} \right) \]

This simplifies to:

\[ \frac{1}{10} = 0.5 \left( \frac{2}{R} \right) \]

From this, we can solve for R:

\[ \frac{1}{10} = \frac{1}{R} \]
\[ R = 10 \text{ cm} \]

### Step 3: Find the focal length of the mirror
When one face of the lens is silvered, it behaves like a mirror. The focal length of a spherical mirror is given by:

\[ f_{mirror} = \frac{R}{2} \]

So, substituting R:

\[ f_{mirror} = \frac{10}{2} = 5 \text{ cm} \]

### Step 4: Use the mirror formula
The mirror formula is given by:

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

Given that the image coincides with the object, we have \( v = -u \) (the negative sign indicates that the image is on the same side as the object for a mirror). Substituting this into the mirror formula:

\[ \frac{1}{-u} + \frac{1}{u} = \frac{1}{-5/2} \]

This simplifies to:

\[ -\frac{1}{u} + \frac{1}{u} = -\frac{2}{5} \]

This leads to:

\[ -\frac{2}{u} = -\frac{2}{5} \]

### Step 5: Solve for u
Cross-multiplying gives:

\[ 2 \cdot 5 = 2u \]
\[ 10 = 2u \]
\[ u = 5 \text{ cm} \]

### Final Answer
The value of \( u \) is **5 cm**.

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