A concave spherical mirror with a radius of curvature of `0.2 m` is filled with water. What is the focal length of this system? Refractive index of water is `4//3.`

Explanation

To find the focal length of a concave spherical mirror filled with water, we can follow these steps:

### Step 1: Identify the given values
- Radius of curvature of the concave mirror, \( R = -0.2 \, \text{m} \) (negative because it is a concave mirror)
- Refractive index of air, \( \mu_1 = 1 \)
- Refractive index of water, \( \mu_2 = \frac{4}{3} \)

### Step 2: Convert the radius of curvature to centimeters
Since we often work in centimeters in optics, convert the radius of curvature:
\[ R = -0.2 \, \text{m} = -20 \, \text{cm} \]

### Step 3: Use the formula for equivalent focal length
The formula for the equivalent focal length \( f \) of a system involving a concave mirror and a medium with a different refractive index is given by:
\[ \frac{1}{f} = \frac{2 \mu_2}{\mu_1 R} - \frac{2 \mu_2}{\mu_1} \cdot \frac{1}{R_1} \]
Where:
- \( R_1 \) is the radius of curvature of the plane surface (which is considered to be infinity for a plane surface, thus \( \frac{1}{R_1} = 0 \)).

### Step 4: Substitute the values into the formula
Substituting the values into the formula:
\[ \frac{1}{f} = \frac{2 \cdot \frac{4}{3}}{1 \cdot (-20)} - 0 \]
This simplifies to:
\[ \frac{1}{f} = \frac{8/3}{-20} \]

### Step 5: Simplify the expression
Calculating the right-hand side:
\[ \frac{1}{f} = \frac{8}{3 \cdot -20} = \frac{8}{-60} = -\frac{2}{15} \]

### Step 6: Find the focal length
Taking the reciprocal to find \( f \):
\[ f = -\frac{15}{2} = -7.5 \, \text{cm} \]

### Conclusion
The focal length of the concave mirror filled with water is \( -7.5 \, \text{cm} \). The negative sign indicates that the system behaves like a concave mirror.

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