The least distance between a point object and its real image formed by a convex lens of focal length F is

Options

A. 2 F
B. 3 F
C. 4 F
D. Greater than 4 F

Answer

To find the least distance between a point object and its real image formed by a convex lens of focal length \( F \), we can follow these steps:

### Step 1: Understand the Lens Formula
The lens formula for a convex lens 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 (positive for real images),
- \( u \) is the object distance (negative for real objects).

### Step 2: Define the Distances
Let:
- \( d \) be the least distance between the object and the image,
- \( x \) be the distance of the image from the lens.

Then, the object distance \( u \) can be expressed as:
\[ u = -(d - x) \]
This is because the object is located at a distance \( d - x \) to the left of the lens.

### Step 3: Substitute into the Lens Formula
Substituting \( u \) and \( v \) into the lens formula:
\[ \frac{1}{F} = \frac{1}{x} - \frac{1}{-(d-x)} \]
This simplifies to:
\[ \frac{1}{F} = \frac{1}{x} + \frac{1}{d-x} \]

### Step 4: Combine the Fractions
Combining the fractions on the right-hand side:
\[ \frac{1}{F} = \frac{(d-x) + x}{x(d-x)} = \frac{d}{x(d-x)} \]

### Step 5: Rearranging the Equation
Rearranging gives us:
\[ d = \frac{F \cdot x(d-x)}{1} \]

### Step 6: Find the Minimum Distance
To find the least distance \( d \), we can take the derivative of \( d \) with respect to \( x \) and set it to zero to find the critical points. However, we can also use the fact that the least distance occurs when the object and image distances are equal (i.e., \( u = -v \)).

Setting \( d = 2x \):
\[ d = 2x \implies x = \frac{d}{2} \]

Substituting \( x \) back into the equation gives:
\[ d = 4F \]

### Conclusion
Thus, the least distance \( d \) between the point object and its real image formed by a convex lens of focal length \( F \) is:
\[ \boxed{4F} \]