A convex lens of focal length 16 cm forms a real image double the size of the object. The image distance of the object from the lens is

Options

A. 32 cm
B. 24 cm
C. 20 cm
D. 8 cm

Explanation

To find the image distance of the object from the lens, we can follow these steps:

### Step 1: Understand the given information
We have a convex lens with a focal length (f) of 16 cm. The magnification (m) is given as -2 because the image is real and inverted, and it is double the size of the object.

### Step 2: Use the magnification formula
The magnification (m) is defined as:
\[ m = \frac{h'}{h} = -\frac{v}{u} \]
Where:
- \( h' \) is the height of the image,
- \( h \) is the height of the object,
- \( v \) is the image distance,
- \( u \) is the object distance.

Given that \( m = -2 \):
\[ -2 = -\frac{v}{u} \]
This simplifies to:
\[ v = 2u \]

### Step 3: Use the lens formula
The lens formula relates the focal length (f), object distance (u), and image distance (v):
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]

Substituting \( f = 16 \) cm and \( v = 2u \):
\[ \frac{1}{16} = \frac{1}{2u} - \frac{1}{u} \]

### Step 4: Simplify the equation
To simplify the right side, we can find a common denominator:
\[ \frac{1}{16} = \frac{1}{2u} - \frac{2}{2u} \]
\[ \frac{1}{16} = -\frac{1}{2u} \]

### Step 5: Solve for u
Now, we can rearrange the equation to solve for \( u \):
\[ -\frac{1}{2u} = \frac{1}{16} \]
Taking the reciprocal of both sides gives:
\[ -2u = 16 \]
Thus,
\[ u = -8 \text{ cm} \]

### Step 6: Find the image distance (v)
Now substituting \( u \) back into the equation \( v = 2u \):
\[ v = 2(-8) = -16 \text{ cm} \]

### Conclusion
The image distance of the object from the lens is \( v = -16 \) cm.

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