Two lenses are placed in contact with each other and the focal length of combination is `80 cm`. If the focal length of one is `20 cm`, then the power of the other will be

Options

A. 1.66 D
B. 4.00 D
C. `-1.00 D`
D. `-3.75 D`

Explanation

To solve the problem step by step, we will use the lens formula and the concept of power of lenses.

### Step 1: Understand the Given Information
We have two lenses in contact:
- Focal length of the combination (F_combination) = 80 cm
- Focal length of the first lens (F1) = 20 cm

### Step 2: Use the Formula for Focal Length of Combination
The formula for the focal length of two lenses in contact is given by:
\[ \frac{1}{F_{\text{combination}}} = \frac{1}{F_1} + \frac{1}{F_2} \]
where \(F_2\) is the focal length of the second lens.

### Step 3: Substitute the Known Values
Substituting the known values into the equation:
\[ \frac{1}{80} = \frac{1}{20} + \frac{1}{F_2} \]

### Step 4: Solve for \(F_2\)
Rearranging the equation to solve for \(\frac{1}{F_2}\):
\[ \frac{1}{F_2} = \frac{1}{80} - \frac{1}{20} \]
Finding a common denominator (which is 80):
\[ \frac{1}{F_2} = \frac{1}{80} - \frac{4}{80} = \frac{-3}{80} \]
Thus,
\[ F_2 = -\frac{80}{3} \text{ cm} \approx -26.67 \text{ cm} \]

### Step 5: Calculate the Power of the Second Lens
The power \(P\) of a lens is given by the formula:
\[ P = \frac{100}{F} \text{ (in cm)} \]
For the second lens:
\[ P_2 = \frac{100}{F_2} = \frac{100}{-\frac{80}{3}} = -\frac{300}{80} = -3.75 \text{ diopters} \]

### Step 6: Conclusion
The power of the second lens is \(-3.75\) diopters.

---