A double convex lens has two surfaces of equal radii R and refractive index m=1.5, we have

Options

A. `f=R/2`
B. `f=R`
C. `f=-R`
D. `f=2R`

Answer

To solve the problem regarding a double convex lens with equal radii \( R \) and a refractive index \( \mu = 1.5 \), we will use the lens maker's formula. The lens maker's formula is given by:

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

### Step 1: Identify the radii of curvature
For a double convex lens:
- The radius of curvature of the first surface, \( R_1 \), is positive: \( R_1 = +R \)
- The radius of curvature of the second surface, \( R_2 \), is negative: \( R_2 = -R \)

### Step 2: Substitute the values into the lens maker's formula
Substituting \( R_1 \) and \( R_2 \) into the lens maker's formula:

\[ \frac{1}{f} = \left( \mu - 1 \right) \left( \frac{1}{R} - \frac{1}{-R} \right) \]

### Step 3: Simplify the equation
Calculating \( \frac{1}{R} - \frac{1}{-R} \):

\[ \frac{1}{R} - \frac{1}{-R} = \frac{1}{R} + \frac{1}{R} = \frac{2}{R} \]

Now substituting this back into the equation:

\[ \frac{1}{f} = \left( 1.5 - 1 \right) \left( \frac{2}{R} \right) \]

### Step 4: Calculate the focal length
Now, simplifying further:

\[ \frac{1}{f} = 0.5 \cdot \frac{2}{R} = \frac{1}{R} \]

Thus, we find:

\[ f = R \]

### Conclusion
The focal length \( f \) of the double convex lens is equal to the radius of curvature \( R \).

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