A convex lens of focal length f produces an image `(1)/(n)` times than that of the size of the object. The distance of the object from the lens is

Options

A. nf
B. `(f)/(n)`
C. (n + 1) f
D. (n - 1) f

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We have a convex lens with a focal length \( f \). The image produced by the lens is \( \frac{1}{n} \) times the size of the object. This means the magnification \( m \) is given by:

\[ m = \frac{h'}{h} = \frac{1}{n} \]

where \( h' \) is the height of the image and \( h \) is the height of the object.

### Step 2: Relate magnification to object and image distances
The magnification \( m \) can also be expressed in terms of the object distance \( u \) and the image distance \( v \):

\[ m = \frac{v}{u} \]

From the magnification formula, we can write:

\[ \frac{v}{u} = \frac{1}{n} \]

This implies:

\[ v = \frac{u}{n} \]

### Step 3: Use the lens formula
The lens formula for a convex lens is given by:

\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]

Substituting the expression for \( v \) from Step 2 into the lens formula:

\[ \frac{1}{f} = \frac{1}{\frac{u}{n}} - \frac{1}{u} \]

### Step 4: Simplify the equation
The left side becomes:

\[ \frac{1}{f} = \frac{n}{u} - \frac{1}{u} \]

Combining the terms on the right side gives:

\[ \frac{1}{f} = \frac{n - 1}{u} \]

### Step 5: Rearranging to find \( u \)
Now, we can rearrange this equation to solve for \( u \):

\[ u = (n - 1)f \]

### Step 6: Find the final expression for the object distance
To express \( u \) in terms of \( f \) and \( n \):

\[ u = n f + f \]

Thus, the final expression for the distance of the object from the lens is:

\[ u = \frac{n f}{n - 1} \]

### Summary
The distance of the object from the lens is given by:

\[ u = \frac{n f}{n - 1} \]