To find the power of the double convex lens, we will follow these steps:
### Step 1: Identify the radii of curvature
The radii of curvature for the double convex lens are given as:
- \( R_1 = 10 \, \text{cm} \) (for the first surface)
- \( R_2 = 20 \, \text{cm} \) (for the second surface)
### Step 2: Determine the sign of the radii
In the lens maker's formula, the convention is that:
- The radius of curvature \( R_1 \) is positive for a convex surface when light is incident from the left.
- The radius of curvature \( R_2 \) is negative for a convex surface when light is incident from the left.
Thus, we have:
- \( R_1 = +10 \, \text{cm} \)
- \( R_2 = -20 \, \text{cm} \)
### Step 3: Use the lens maker's formula
The lens maker's formula is given by:
\[
\frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)
\]
where:
- \( n \) is the refractive index of the lens material,
- \( f \) is the focal length of the lens.
Given that the refractive index \( n = 1.5 \), we can substitute the values into the formula:
\[
\frac{1}{f} = (1.5 - 1) \left( \frac{1}{10} - \frac{1}{-20} \right)
\]
\[
= 0.5 \left( \frac{1}{10} + \frac{1}{20} \right)
\]
### Step 4: Calculate the terms inside the parentheses
To calculate \( \frac{1}{10} + \frac{1}{20} \):
\[
\frac{1}{10} = \frac{2}{20}
\]
Thus,
\[
\frac{1}{10} + \frac{1}{20} = \frac{2}{20} + \frac{1}{20} = \frac{3}{20}
\]
### Step 5: Substitute back into the formula
Now substituting back:
\[
\frac{1}{f} = 0.5 \left( \frac{3}{20} \right) = \frac{3}{40}
\]
### Step 6: Find the focal length \( f \)
Taking the reciprocal to find \( f \):
\[
f = \frac{40}{3} \, \text{cm}
\]
### Step 7: Convert focal length to meters
To convert \( f \) from centimeters to meters:
\[
f = \frac{40}{3} \, \text{cm} = \frac{40}{3 \times 100} \, \text{m} = \frac{40}{300} \, \text{m} = \frac{2}{15} \, \text{m}
\]
### Step 8: Calculate the power of the lens
The power \( P \) of the lens in diopters is given by:
\[
P = \frac{1}{f} \, (\text{in meters})
\]
Thus,
\[
P = \frac{1}{\frac{2}{15}} = \frac{15}{2} = 7.5 \, \text{diopters}
\]
### Final Answer
The power of the lens is \( +7.5 \, \text{D} \).
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