In a plano-convex lens radius of curvature of the lens is 10 cm. if the plane side polished, then the magnitude of the focal length of the mirror so formed will be (refractive index =1.5) (2x) cm . Find value of x.

Answer

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

### Step 1: Understand the given information
We have a plano-convex lens with a radius of curvature (R) of 10 cm and a refractive index (μ) of 1.5. The plane side of the lens is polished, and we need to find the focal length of the mirror formed, which is given as (2x) cm.

### Step 2: Identify the focal length of the plano-convex lens
For a plano-convex lens, the formula for the focal length (f) is given by:

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

Where:
- \( R_1 \) is the radius of curvature of the convex side (10 cm),
- \( R_2 \) is the radius of curvature of the plane side (which is considered to be infinite, so \( R_2 = \infty \)).

### Step 3: Substitute the values into the formula
Since \( R_2 \) is infinite:

\[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{10} - 0 \right) \]

This simplifies to:

\[ \frac{1}{f} = 0.5 \cdot \frac{1}{10} \]

### Step 4: Calculate the focal length
Now, calculate \( f \):

\[ \frac{1}{f} = \frac{0.5}{10} = \frac{0.05}{1} \]

Thus,

\[ f = \frac{1}{0.05} = 20 \text{ cm} \]

### Step 5: Relate the focal length of the mirror to the lens
When the plane side is polished, the focal length of the mirror (F) formed is given by:

\[ F = 2f \]

Substituting the value of \( f \):

\[ F = 2 \times 20 = 40 \text{ cm} \]

### Step 6: Relate to the problem statement
According to the problem, the focal length of the mirror is also given as \( 2x \):

\[ 2x = 40 \]

### Step 7: Solve for x
Now, divide both sides by 2:

\[ x = \frac{40}{2} = 20 \text{ cm} \]

### Final Answer
The value of \( x \) is 20 cm.

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