A convex lens of focal length 40 cm is in contact with a concave lens of focal length 25 cm. The power of the combination is

Options

A. `-1.25D`
B. `-6.5D`
C. `+6.5D`
D. `+1.25D`

Explanation

To find the power of the combination of a convex lens and a concave lens in contact, we can follow these steps:

### Step 1: Identify the Focal Lengths
- The focal length of the convex lens (F1) is given as +40 cm.
- The focal length of the concave lens (F2) is given as -25 cm (since concave lenses have negative focal lengths).

### Step 2: Use the Formula for Combined Focal Length
The formula for the combined focal length (F) of two lenses in contact is:
\[ \frac{1}{F} = \frac{1}{F_1} + \frac{1}{F_2} \]
Substituting the values:
\[ \frac{1}{F} = \frac{1}{40} + \frac{1}{-25} \]

### Step 3: Calculate the Individual Terms
Convert the fractions to a common denominator:
- The common denominator of 40 and 25 is 200.
- Thus, we rewrite the fractions:
\[ \frac{1}{40} = \frac{5}{200} \quad \text{and} \quad \frac{1}{-25} = \frac{-8}{200} \]
Now substituting these values back into the equation:
\[ \frac{1}{F} = \frac{5}{200} - \frac{8}{200} = \frac{-3}{200} \]

### Step 4: Solve for Focal Length F
Taking the reciprocal gives:
\[ F = \frac{-200}{3} \text{ cm} \]

### Step 5: Calculate the Power of the Lens Combination
The power (P) of a lens is given by the formula:
\[ P = \frac{1}{F} \text{ (in meters)} \]
Since our focal length is in centimeters, we convert it to meters by using the relation:
\[ P = \frac{100}{F} \text{ (when F is in cm)} \]
Substituting the value of F:
\[ P = \frac{100}{\left(-\frac{200}{3}\right)} = \frac{100 \times 3}{-200} = -1.5 \text{ diopters} \]

### Conclusion
The power of the combination of the convex lens and the concave lens is:
\[ \text{Power} = -1.5 \text{ diopters} \]

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