A farsighted person cannot see objects placed closer to 50 cm. Find the power of the lens needed to see the objects at 20 cm.

Answer

To solve the problem of finding the power of the lens needed for a farsighted person who cannot see objects closer than 50 cm and needs to see objects at 20 cm, we can follow these steps:

### Step 1: Understand the Given Values
- The farthest distance a farsighted person can see clearly (the near point) is given as \( D = 50 \, \text{cm} \).
- The distance at which the person wants to see clearly (the object distance) is \( u = -20 \, \text{cm} \) (the negative sign indicates that the object is on the same side as the incoming light).

### Step 2: Use the Lens Formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
where:
- \( f \) is the focal length of the lens,
- \( v \) is the image distance (which is equal to the near point distance for a farsighted person, so \( v = -50 \, \text{cm} \)),
- \( u \) is the object distance (which is \( -20 \, \text{cm} \)).

### Step 3: Substitute the Values into the Lens Formula
Substituting the values into the lens formula:
\[ \frac{1}{f} = \frac{1}{-50} - \frac{1}{-20} \]
This simplifies to:
\[ \frac{1}{f} = -\frac{1}{50} + \frac{1}{20} \]

### Step 4: Find a Common Denominator and Simplify
The common denominator for 50 and 20 is 100. Thus, we can rewrite the fractions:
\[ \frac{1}{f} = -\frac{2}{100} + \frac{5}{100} = \frac{3}{100} \]

### Step 5: Calculate the Focal Length
Now, we can find \( f \):
\[ f = \frac{100}{3} \, \text{cm} \]

### Step 6: Convert Focal Length to Meters
To convert the focal length into meters:
\[ f = \frac{100}{3} \, \text{cm} = \frac{100}{3 \times 100} \, \text{m} = \frac{1}{3} \, \text{m} \]

### Step 7: Calculate the Power of the Lens
The power \( P \) of the lens is given by:
\[ P = \frac{1}{f} \, \text{(in meters)} \]
Substituting the value of \( f \):
\[ P = \frac{1}{\frac{1}{3}} = 3 \, \text{diopters} \]

### Final Answer
The power of the lens needed for the farsighted person to see objects at 20 cm is **3 diopters**.

---