Convex lens made up of glass `(mu_(g)=1.5)` and radius of curvature R is dipped into water. Its focal length will be (Refractice index of water =4/3)

Options

A. 4R
B. 2R
C. R
D. `(R)/(2)`

Answer

To find the focal length of a convex lens made of glass when it is dipped in water, we will use the lens maker's formula. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Lens Maker's Formula**:
The lens maker's formula is given by:
\[ \frac{1}{f} = \left(\frac{\mu_L}{\mu_M} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \]
where:
- \( f \) is the focal length of the lens.
- \( \mu_L \) is the refractive index of the lens material (glass).
- \( \mu_M \) is the refractive index of the medium (water).
- \( R_1 \) and \( R_2 \) are the radii of curvature of the lens surfaces.

2. **Identifying the Given Values**:
- Refractive index of glass, \( \mu_L = 1.5 \)
- Refractive index of water, \( \mu_M = \frac{4}{3} \)
- Radius of curvature, \( R = r \) (we will assume both surfaces have the same radius for simplicity).

3. **Assigning Radii of Curvature**:
For a convex lens:
- The first surface (convex) has a radius of curvature \( R_1 = r \) (positive).
- The second surface (concave) has a radius of curvature \( R_2 = -r \) (negative).

4. **Substituting Values into the Formula**:
Plugging in the values into the lens maker's formula:
\[ \frac{1}{f} = \left(\frac{1.5}{\frac{4}{3}} - 1\right) \left(\frac{1}{r} - \left(-\frac{1}{r}\right)\right) \]

5. **Calculating the Refractive Index Ratio**:
First, calculate the ratio:
\[ \frac{1.5}{\frac{4}{3}} = \frac{1.5 \times 3}{4} = \frac{4.5}{4} = \frac{9}{8} \]
Therefore:
\[ \frac{1}{f} = \left(\frac{9}{8} - 1\right) \left(\frac{1}{r} + \frac{1}{r}\right) \]
Simplifying further:
\[ \frac{1}{f} = \left(\frac{9}{8} - \frac{8}{8}\right) \left(\frac{2}{r}\right) = \left(\frac{1}{8}\right) \left(\frac{2}{r}\right) \]

6. **Final Calculation**:
\[ \frac{1}{f} = \frac{2}{8r} = \frac{1}{4r} \]
Therefore, the focal length \( f \) is:
\[ f = 4r \]

### Conclusion:
The focal length of the convex lens when dipped in water is \( 4r \).