An equiconvex lens of glass `(mu_(g) = 1.5)` of focal length 10 cm is silvered on one side. It will behave like a

Options

A. concave mirror of focal length 10 cm
B. convex mirror of focal length 5.0 cm
C. concave mirror of focal length 2.5 cm
D. convex mirror of focal length 20cm

Answer

To solve the problem of determining how an equiconvex lens of glass (with a refractive index \( \mu_g = 1.5 \) and a focal length of 10 cm) behaves when silvered on one side, we can follow these steps:

### Step 1: Understand the System
The given system consists of an equiconvex lens that is silvered on one side. When one side of the lens is silvered, it behaves like a combination of a lens and a mirror.

### Step 2: Identify the Focal Length of the Lens
The focal length of the equiconvex lens is given as \( f_L = 10 \, \text{cm} \).

### Step 3: Calculate the Focal Length of the Mirror
For a mirror, the focal length \( f_M \) is related to the radius of curvature \( R \) by the formula:
\[ f_M = \frac{R}{2} \]
To find \( R \), we can use the lens maker's formula for the lens:
\[ \frac{1}{f_L} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
For an equiconvex lens, \( R_1 = R \) and \( R_2 = -R \). Thus:
\[ \frac{1}{f_L} = (1.5 - 1) \left( \frac{1}{R} - \left(-\frac{1}{R}\right) \right) = \frac{0.5 \times 2}{R} = \frac{1}{R} \]
From this, we can find \( R \):
\[ R = f_L = 10 \, \text{cm} \]
Now, substituting \( R \) into the mirror focal length formula:
\[ f_M = \frac{10}{2} = 5 \, \text{cm} \]

### Step 4: Combine the Focal Lengths
When the lens is silvered on one side, the effective focal length \( f_{comb} \) of the combination can be calculated using the formula:
\[ \frac{1}{f_{comb}} = \frac{1}{f_L} + \frac{1}{f_M} \]
Substituting the values:
\[ \frac{1}{f_{comb}} = \frac{1}{10} + \frac{1}{5} = \frac{1}{10} + \frac{2}{10} = \frac{3}{10} \]
Thus, the focal length of the combination is:
\[ f_{comb} = \frac{10}{3} \approx 3.33 \, \text{cm} \]

### Step 5: Determine the Sign of the Focal Length
Since the rays converge in front of the lens-mirror combination, the focal length is considered negative:
\[ f_{comb} = -3.33 \, \text{cm} \]

### Conclusion
The system behaves like a concave mirror with a focal length of approximately \( -3.33 \, \text{cm} \).