A plano-convex lens P and a concavo-convex lens Q are in contact as shown in figure. The refractive index o fthe material of the lens P and Q is `1.8 and 1.2` respectively. The radius of curvature of the concave surface of the lens Q is double the radius of curvature of the convex surface. The convex surface of Q is silvered.The focal lengthof combination is 8cm. The radius of curvature of common surface is

Options

A. `48 cm`
B. `24 cm`
C. `12 cm`
D. `8 cm`

Explanation

To solve the problem step by step, we need to analyze the given information about the plano-convex lens P and the concavo-convex lens Q, and then apply the lens formula and the concept of power of lenses.

### Step 1: Identify the given information
- Refractive index of lens P, \( n_P = 1.8 \)
- Refractive index of lens Q, \( n_Q = 1.2 \)
- Focal length of the combination, \( f = 8 \) cm
- Radius of curvature of the convex surface of lens Q is \( r \)
- Radius of curvature of the concave surface of lens Q is \( 2r \)

### Step 2: Write the lens maker's formula for both lenses
The lens maker's formula is given by:
\[ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
For lens P (plano-convex):
- \( R_1 = +r \) (convex surface)
- \( R_2 = \infty \) (plano surface)

Thus, the formula becomes:
\[ \frac{1}{f_P} = (1.8 - 1) \left( \frac{1}{r} - 0 \right) = 0.8 \cdot \frac{1}{r} \]
So,
\[ f_P = \frac{r}{0.8} = \frac{5r}{4} \]

For lens Q (concavo-convex):
- \( R_1 = -2r \) (concave surface, negative)
- \( R_2 = +r \) (convex surface)

Thus, the formula becomes:
\[ \frac{1}{f_Q} = (1.2 - 1) \left( \frac{1}{-2r} - \frac{1}{r} \right) \]
This simplifies to:
\[ \frac{1}{f_Q} = 0.2 \left( -\frac{1}{2r} - \frac{1}{r} \right) = 0.2 \left( -\frac{1 + 2}{2r} \right) = -\frac{0.2 \cdot 3}{2r} = -\frac{0.3}{r} \]
So,
\[ f_Q = -\frac{r}{0.3} = -\frac{10r}{3} \]

### Step 3: Find the combined focal length
The focal length of the combination of the two lenses in contact is given by:
\[ \frac{1}{f} = \frac{1}{f_P} + \frac{1}{f_Q} \]
Substituting the values we found:
\[ \frac{1}{8} = \frac{4}{5r} - \frac{3}{10r} \]

### Step 4: Solve for \( r \)
Finding a common denominator (which is \( 10r \)):
\[ \frac{1}{8} = \frac{8}{10r} - \frac{3}{10r} = \frac{5}{10r} = \frac{1}{2r} \]
Thus,
\[ 2r = 8 \implies r = 4 \text{ cm} \]

### Step 5: Find the radius of curvature of the common surface
The radius of curvature of the common surface is given by:
\[ R = 2r = 2 \times 4 = 8 \text{ cm} \]

### Final Answer
The radius of curvature of the common surface is \( 8 \) cm.

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