The distance between an object and a divergent lens is m times the focal length of the lens. The linear magnification produced by the lens is

Options

A. m
B. `1/m`
C. m+1
D. `1/(m+1)`

Explanation

To solve the problem of finding the linear magnification produced by a divergent lens when the object distance is m times the focal length, we can follow these steps:

### Step 1: Understand the relationship between object distance (u), image distance (v), and focal length (f)
For a lens, the lens formula is given by:
\[ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \]
where:
- \( u \) is the object distance (negative for a real object),
- \( v \) is the image distance (positive for a virtual image in the case of a divergent lens),
- \( f \) is the focal length (negative for a divergent lens).

### Step 2: Define the object distance in terms of the focal length
Given that the object distance \( u \) is \( m \) times the focal length \( f \):
\[ u = -m f \]
(Note: The negative sign indicates that the object is placed on the same side as the incoming light for a lens.)

### Step 3: Substitute \( u \) into the lens formula
Substituting \( u \) into the lens formula:
\[ \frac{1}{v} - \frac{1}{-m f} = \frac{1}{f} \]
This simplifies to:
\[ \frac{1}{v} + \frac{1}{m f} = \frac{1}{f} \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{m f} \]
\[ \frac{1}{v} = \frac{m - 1}{m f} \]

### Step 5: Find the image distance \( v \)
Taking the reciprocal to find \( v \):
\[ v = \frac{m f}{m - 1} \]

### Step 6: Calculate the magnification
The linear magnification \( M \) produced by the lens is given by:
\[ M = \frac{v}{u} \]
Substituting the values of \( v \) and \( u \):
\[ M = \frac{\frac{m f}{m - 1}}{-m f} \]
This simplifies to:
\[ M = -\frac{1}{m - 1} \]

### Final Answer
Thus, the linear magnification produced by the lens is:
\[ M = -\frac{1}{m - 1} \]