Power `(P)` of a lens is given by reciprocal of focal length `(f)` of the lens. i.e. `P = 1//f`. When `f` is in metre, `P` is in dioptre. For a convex lens, power is positive and for a concave lens, power is negative. When a number of thin lenses of powers `p_(1), p_(2), p_(3)....` are held in contact with one another, the power of the combination is given by algebraic sum of the powers of all the lenses
i.e., `P = p_(1) + p_(2) + p_(3) + ........`
Answer the following questions :
Two thin lenses are in contact and the focal length of the combination is `80 cm`. If the focal length of one lens is `20 cm`, the focal length of the other would be

Options

A. `-26.7 cm`
B. `60 cm`
C. `80 cm`
D. `20 cm`

Explanation

To solve the problem, we will use the formula for the combination of two thin lenses in contact. The formula states that the reciprocal of the focal length of the combination (F) is equal to the sum of the reciprocals of the individual focal lengths (f1 and f2) of the lenses:

\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \]

### Step 1: Identify the given values
- The focal length of the combination, \( F = 80 \, \text{cm} \)
- The focal length of one lens, \( f_1 = 20 \, \text{cm} \)
- We need to find the focal length of the other lens, \( f_2 \).

### Step 2: Write the equation for the combination of lenses
Using the formula mentioned above, we can substitute the known values:

\[ \frac{1}{80} = \frac{1}{20} + \frac{1}{f_2} \]

### Step 3: Rearrange the equation to solve for \( \frac{1}{f_2} \)
First, we can calculate \( \frac{1}{20} \):

\[ \frac{1}{20} = 0.05 \]

Now substituting this into the equation:

\[ \frac{1}{80} = 0.0125 \]

So we have:

\[ 0.0125 = 0.05 + \frac{1}{f_2} \]

### Step 4: Isolate \( \frac{1}{f_2} \)
To isolate \( \frac{1}{f_2} \), we can rearrange the equation:

\[ \frac{1}{f_2} = 0.0125 - 0.05 \]

Calculating the right side:

\[ \frac{1}{f_2} = 0.0125 - 0.05 = -0.0375 \]

### Step 5: Calculate \( f_2 \)
Now, we take the reciprocal to find \( f_2 \):

\[ f_2 = \frac{1}{-0.0375} = -26.67 \, \text{cm} \]

### Conclusion
The focal length of the other lens is approximately \( -26.67 \, \text{cm} \).

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