The radius of curvature of the convex surface of a planoconvex lens is 12 cm and its refractive index 1.5. If the plane face of the lens is silvered, then the distance from the lens at which parallel rays incident on its convex surface converge is

Options

A. 12 cm
B. 18 cm
C. 24 cm
D. 30 cm

Explanation

To solve the problem step by step, we will follow these instructions:

### Step 1: Identify the parameters of the plano-convex lens
- The radius of curvature (R) of the convex surface is given as 12 cm.
- The refractive index (μ) of the lens is given as 1.5.

### Step 2: Calculate the focal length of the plano-convex lens
The formula for the focal length (F) of a lens is given by the lens maker's formula:
\[ \frac{1}{F} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
For a plano-convex lens:
- \( R_1 = +12 \, \text{cm} \) (convex surface, positive)
- \( R_2 = \infty \) (plane surface, negative)

Substituting the values:
\[ \frac{1}{F} = (1.5 - 1) \left( \frac{1}{12} - 0 \right) \]
\[ \frac{1}{F} = 0.5 \cdot \frac{1}{12} = \frac{0.5}{12} = \frac{1}{24} \]
Thus, the focal length \( F \) is:
\[ F = 24 \, \text{cm} \]

### Step 3: Silver the plane face of the lens
When the plane face of the lens is silvered, it behaves like a combination of a lens and a mirror. The focal length of a plane mirror is infinite.

### Step 4: Calculate the focal length of the combination
Using the formula for the focal length of a combination of a lens and a mirror:
\[ \frac{1}{F_{comb}} = \frac{1}{F_{mirror}} - \frac{2}{F_{lens}} \]
Since the focal length of the plane mirror is infinite:
\[ \frac{1}{F_{mirror}} = 0 \]
And the focal length of the plano-convex lens is \( F_{lens} = 24 \, \text{cm} \):
\[ \frac{1}{F_{comb}} = 0 - \frac{2}{24} \]
\[ \frac{1}{F_{comb}} = -\frac{1}{12} \]
Thus, the focal length of the combination is:
\[ F_{comb} = -12 \, \text{cm} \]

### Step 5: Interpret the result
The negative sign indicates that the combination behaves like a concave mirror. The distance from the lens at which parallel rays converge is the absolute value of the focal length:
\[ \text{Distance} = 12 \, \text{cm} \]

### Final Answer
The distance from the lens at which parallel rays incident on its convex surface converge is **12 cm**.

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