A plano-convex lens `(mu = 1.5)` has a curved surface of radiys 15 cm. what is its focal length ?

Answer

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

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

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-by-step Solution:

1. **Identify the parameters:**
- The refractive index \( \mu = 1.5 \).
- The radius of curvature of the curved surface \( R_2 = 15 \, \text{cm} \).
- The plano surface has an infinite radius of curvature, so \( R_1 = \infty \).

2. **Substitute the values into the lens maker's formula:**
- Since \( R_1 = \infty \), we have \( \frac{1}{R_1} = 0 \).
- Therefore, the formula simplifies to:
\[ \frac{1}{F} = \left( 1.5 - 1 \right) \left( 0 - \frac{1}{15} \right) \]

3. **Calculate the expression:**
- First, calculate \( \mu - 1 \):
\[ \mu - 1 = 1.5 - 1 = 0.5 \]
- Now substitute this back into the equation:
\[ \frac{1}{F} = 0.5 \left( -\frac{1}{15} \right) = -\frac{0.5}{15} = -\frac{1}{30} \]

4. **Find the focal length \( F \):**
- Taking the reciprocal gives:
\[ F = -30 \, \text{cm} \]

### Final Answer:
The focal length of the plano-convex lens is \( -30 \, \text{cm} \).