A plano convex lens of radius of curvature 30 cm and refractive index 1.5 is kept in air. Find its focal length (in cm).

Explanation

To find the focal length of a plano-convex lens, we can use the lens maker's formula:

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

Where:
- \( f \) is the focal length of the lens,
- \( \mu \) is the refractive index of the lens material,
- \( R_1 \) is the radius of curvature of the first surface,
- \( R_2 \) is the radius of curvature of the second surface.

### Step 1: Identify the given values
- Refractive index \( \mu = 1.5 \)
- Radius of curvature \( R = 30 \, \text{cm} \) (for the convex surface, we take it as positive)
- For a plano-convex lens, the second surface is flat, so \( R_1 = +30 \, \text{cm} \) and \( R_2 = \infty \) (flat surface).

### Step 2: Substitute the values into the formula
Since \( R_2 \) is flat, we have:
\[ \frac{1}{R_2} = 0 \]
Thus, the formula simplifies to:
\[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{30} - 0 \right) \]

### Step 3: Calculate the right-hand side
\[ \frac{1}{f} = 0.5 \cdot \frac{1}{30} \]
\[ \frac{1}{f} = \frac{0.5}{30} = \frac{1}{60} \]

### Step 4: Find the focal length
Taking the reciprocal gives us:
\[ f = 60 \, \text{cm} \]

Thus, the focal length of the plano-convex lens is **60 cm**.

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