A plano - convex lens has diameter 5 cm and its thickness at the centre is 0.25 cm . If the speed of light inside the lens is `2xx10^(8)ms^(-1)`, then what is the focal length (in cm) of the lens?

Answer

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

### Step 1: Identify the given values
- Diameter of the lens, \( D = 5 \, \text{cm} \)
- Radius of the lens, \( R = \frac{D}{2} = \frac{5}{2} = 2.5 \, \text{cm} \)
- Thickness at the center, \( t = 0.25 \, \text{cm} \)
- Speed of light in the lens, \( v = 2 \times 10^8 \, \text{m/s} \)
- Speed of light in vacuum, \( c = 3 \times 10^8 \, \text{m/s} \)

### Step 2: Calculate the radius of curvature
For a plano-convex lens, the radius of curvature \( R \) can be calculated using the Pythagorean theorem. The radius of curvature is the hypotenuse of a right triangle formed by the thickness and the radius.

Let \( R' \) be the radius of curvature of the convex side of the lens. We can set up the equation:
\[ R'^2 = (2.5)^2 + (R' - 0.25)^2 \]

Expanding this:
\[ R'^2 = 6.25 + (R' - 0.25)^2 \]
\[ R'^2 = 6.25 + (R'^2 - 0.5R' + 0.0625) \]
\[ 0 = 6.25 + 0.0625 - 0.5R' \]
\[ 0 = 6.3125 - 0.5R' \]
\[ 0.5R' = 6.3125 \]
\[ R' = \frac{6.3125}{0.5} = 12.625 \, \text{cm} \]

### Step 3: Calculate the refractive index \( \mu \)
The refractive index \( \mu \) can be calculated using the formula:
\[ \mu = \frac{c}{v} \]
Substituting the values:
\[ \mu = \frac{3 \times 10^8}{2 \times 10^8} = 1.5 \]

### Step 4: Calculate the focal length \( f \)
Using the lens maker's formula for a plano-convex lens:
\[ \frac{1}{f} = \frac{\mu - 1}{R'} \]
Substituting the values:
\[ \frac{1}{f} = \frac{1.5 - 1}{12.625} \]
\[ \frac{1}{f} = \frac{0.5}{12.625} \]
\[ f = \frac{12.625}{0.5} = 25.25 \, \text{cm} \]

### Final Answer
The focal length of the lens is \( \boxed{25.25 \, \text{cm}} \).

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