Diameter of a plano-convex lens is 6cm and thickness at the centre is 3mm. If speed of light in material of lens is `2xx10^(8)(m)/(s)`, The focal length of the lens is

Options

A. 20 cm
B. 30 cm
C. 10 cm
D. 15 cm

Answer

To find the focal length of a plano-convex lens, we can follow these steps:

### Step 1: Calculate the refractive index (μ) of the lens material
The refractive index is given by the formula:
\[ \mu = \frac{c}{v} \]
where \(c\) is the speed of light in vacuum (approximately \(3 \times 10^8 \, m/s\)) and \(v\) is the speed of light in the lens material.

Given:
- \(v = 2 \times 10^8 \, m/s\)

Substituting the values:
\[ \mu = \frac{3 \times 10^8}{2 \times 10^8} = 1.5 \]

### Step 2: Use the lens maker's formula
For a plano-convex lens, the lens maker's formula is given by:
\[ \frac{1}{f} = (\mu - 1) \cdot \frac{1}{R} \]
where \(f\) is the focal length and \(R\) is the radius of curvature of the convex surface.

Substituting the value of \(\mu\):
\[ \frac{1}{f} = (1.5 - 1) \cdot \frac{1}{R} = 0.5 \cdot \frac{1}{R} \]

### Step 3: Calculate the radius of curvature (R)
The diameter of the lens is given as 6 cm, hence the radius (R) is:
\[ R = \frac{6 \, cm}{2} = 3 \, cm \]

However, we need to find the radius of curvature of the convex surface. Since the thickness at the center is 3 mm (which is 0.3 cm), we can use the geometry of the lens to find \(R\).

Using the Pythagorean theorem:
\[ R^2 = (R - 3)^2 + 3^2 \]
Expanding and simplifying:
\[ R^2 = (R^2 - 6R + 9) + 9 \]
\[ R^2 = R^2 - 6R + 18 \]
\[ 6R = 18 \]
\[ R = 3 \, cm + 3 \, cm = 6 \, cm \]

### Step 4: Substitute R back into the lens maker's formula
Now substituting \(R\) back into the lens maker's formula:
\[ \frac{1}{f} = 0.5 \cdot \frac{1}{6} \]
\[ \frac{1}{f} = \frac{0.5}{6} = \frac{1}{12} \]
Thus,
\[ f = 12 \, cm \]

### Final Answer
The focal length of the plano-convex lens is \(12 \, cm\).

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