A divergent lens produces an image of magnification 0.5 when the object distance is 10 cm. The focal power of the lens (in diopters)

Options

A. `+4`
B. `-4`
C. `+10`
D. `-10`

Explanation

To solve the problem, we need to find the focal power of a divergent lens given the magnification and object distance. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Magnification (m) = 0.5
- Object distance (u) = -10 cm (negative because it is a divergent lens)

### 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

### Step 3: Rearrange the formula to find \( v \)
From the magnification formula, we can express \( v \) as:
\[ v = m \cdot u \]
Substituting the values:
\[ v = 0.5 \cdot (-10) \]
\[ v = -5 \, \text{cm} \]

### 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

### Step 5: Substitute the values into the lens formula
Substituting \( v = -5 \, \text{cm} \) and \( u = -10 \, \text{cm} \):
\[ \frac{1}{f} = \frac{1}{-5} - \frac{1}{-10} \]

### Step 6: Simplify the equation
Calculating each term:
\[ \frac{1}{f} = -\frac{1}{5} + \frac{1}{10} \]
Finding a common denominator (10):
\[ \frac{1}{f} = -\frac{2}{10} + \frac{1}{10} \]
\[ \frac{1}{f} = -\frac{1}{10} \]

### Step 7: Solve for \( f \)
Taking the reciprocal gives:
\[ f = -10 \, \text{cm} \]

### Step 8: Calculate the focal power
Focal power (P) is given by:
\[ P = \frac{1}{f(\text{in meters})} \]
Converting \( f \) from cm to meters:
\[ f = -10 \, \text{cm} = -0.1 \, \text{m} \]
Thus,
\[ P = \frac{1}{-0.1} = -10 \, \text{D} \]

### Final Answer
The focal power of the lens is \( -10 \, \text{D} \).

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