A 5.0 diopter lens forms a virtual image which is 4 times the object placed perpendicularly on the principal axis of the lens. Find the distance of the object from the lens.

Explanation

To solve the problem step by step, we will use the lens formula and the magnification formula.

### Step 1: Understand the given information
- The power of the lens (P) = 5.0 diopters
- The magnification (m) = -4 (since the image is virtual and upright)
- We need to find the object distance (u).

### Step 2: Calculate the focal length (f) of the lens
The focal length (f) can be calculated using the formula:
\[ f = \frac{1}{P} \]
Given that P = 5.0 diopters, we convert this to focal length in centimeters:
\[ f = \frac{100}{5} = 20 \text{ cm} \]

### Step 3: Relate the magnification to object and image distances
The magnification (m) is given by the formula:
\[ m = \frac{v}{u} \]
Where:
- \( v \) = image distance
- \( u \) = object distance

Since the magnification is -4, we can write:
\[ -4 = \frac{v}{u} \]
This implies:
\[ v = -4u \]

### Step 4: Use the lens formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Substituting the values we have:
\[ \frac{1}{20} = \frac{1}{-4u} - \frac{1}{u} \]

### Step 5: Simplify the equation
We can rewrite the right side:
\[ \frac{1}{20} = -\frac{1}{4u} - \frac{1}{u} \]
To combine the fractions on the right, we find a common denominator:
\[ \frac{1}{20} = -\frac{1}{4u} - \frac{4}{4u} = -\frac{5}{4u} \]
Now we have:
\[ \frac{1}{20} = -\frac{5}{4u} \]

### Step 6: Cross-multiply to solve for u
Cross-multiplying gives:
\[ 1 \cdot 4u = -5 \cdot 20 \]
This simplifies to:
\[ 4u = -100 \]
Therefore:
\[ u = -\frac{100}{4} = -25 \text{ cm} \]

### Step 7: Conclusion
The distance of the object from the lens is \( u = -25 \) cm. The negative sign indicates that the object is placed on the same side as the incoming light, which is typical for real objects.