If `R_(1)` and `R_(2)` are the radii of curvature of double convex lens made of same material, the lens with more focal length is

Options

A. `R_(1)=20cm,R_(2)=10cm`
B. `R_(1)=R_(2)=20cm`
C. `R_(1)=R_(2)=10cm`
D. `R_(1)=R_(2)=5cm`

Explanation

To solve the problem of determining which double convex lens has a greater focal length based on the radii of curvature \( R_1 \) and \( R_2 \), we can follow these steps:

### Step 1: Understand the Lens Maker's Formula
The focal length \( f \) of a lens is given by the Lens Maker's Formula:
\[ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
where \( n \) is the refractive index of the lens material, and \( R_1 \) and \( R_2 \) are the radii of curvature of the lens surfaces.

### Step 2: Analyze the Formula
For a double convex lens:
- \( R_1 \) is positive (the radius of curvature of the first surface).
- \( R_2 \) is negative (the radius of curvature of the second surface) since it is convex from both sides.

Thus, the formula can be rewritten as:
\[ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} + \frac{1}{|R_2|} \right) \]
where \( |R_2| \) is the absolute value of \( R_2 \).

### Step 3: Determine Conditions for Maximum Focal Length
To maximize the focal length \( f \), we need to minimize \( \frac{1}{f} \). This occurs when both \( R_1 \) and \( |R_2| \) are maximized. Therefore, the focal length will be maximum when both radii of curvature are the largest possible values.

### Step 4: Compare Given Values
If we are given specific values for \( R_1 \) and \( R_2 \), we can directly compare them. For instance, if \( R_1 = 20 \, \text{cm} \) and \( R_2 = 20 \, \text{cm} \), then both radii are equal and at their maximum.

### Step 5: Conclusion
Thus, the lens with the larger focal length is the one with the maximum values of \( R_1 \) and \( R_2 \). If both radii are equal and maximized, the focal length will also be maximized.

### Final Answer
The lens with the more focal length is the one where both \( R_1 \) and \( R_2 \) are maximized, for example, when \( R_1 = 20 \, \text{cm} \) and \( R_2 = 20 \, \text{cm} \).

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