A convex lens is used to obtain a magnified image of an object on a screen . The object is at a distance 10 m from the lens . If the magnification is 19. the focal length of the lens is

Options

A. 9.5 cm
B. `0.95 ` cm
C. 9.5 m
D. 0 . 95 m

Answer

To find the focal length of the convex lens given the object distance and magnification, we can follow these steps:

### Step 1: Understand the given values
- Object distance (U) = -10 m (we take it negative as per the sign convention for lenses)
- Magnification (M) = 19

### Step 2: Use the magnification formula
The magnification (M) is given by the formula:
\[ M = \frac{V}{U} \]
Where:
- V = image distance
- U = object distance

Substituting the known values:
\[ 19 = \frac{V}{-10} \]

### Step 3: Solve for V
Rearranging the equation to find V:
\[ V = 19 \times (-10) \]
\[ V = -190 \text{ m} \]

### Step 4: Use the lens formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{V} - \frac{1}{U} \]
Where:
- f = focal length

Substituting the values of V and U:
\[ \frac{1}{f} = \frac{1}{-190} - \frac{1}{-10} \]

### Step 5: Simplify the equation
Calculating the right side:
\[ \frac{1}{f} = -\frac{1}{190} + \frac{1}{10} \]
To combine these fractions, we need a common denominator:
The common denominator of 190 and 10 is 1900.
So we rewrite the fractions:
\[ \frac{1}{f} = -\frac{10}{1900} + \frac{190}{1900} \]
\[ \frac{1}{f} = \frac{190 - 10}{1900} \]
\[ \frac{1}{f} = \frac{180}{1900} \]

### Step 6: Solve for f
Taking the reciprocal to find f:
\[ f = \frac{1900}{180} \]
\[ f = \frac{190}{18} \]
\[ f \approx 10.56 \text{ m} \]

### Step 7: Convert to centimeters
Since the question may require the answer in centimeters:
\[ f \approx 1056 \text{ cm} \]

### Final Answer
The focal length of the lens is approximately 10.56 m or 1056 cm.