A convex lens of focal length 30 cm produces 5 times magnified real image of an object. What is the object distance?

Options

A. 36 cm
B. 25 cm
C. 30 cm
D. 150 cm

Answer

To find the object distance for a convex lens that produces a 5 times magnified real image, we can follow these steps:

### Step 1: Write down the lens formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
where:
- \( f \) is the focal length of the lens,
- \( v \) is the image distance,
- \( u \) is the object distance.

### Step 2: Identify the given values
From the problem, we know:
- The focal length \( f = 30 \, \text{cm} \) (for a convex lens, this is positive),
- The magnification \( m = 5 \) (since it is a real image, we take it as negative in the lens formula context).

### Step 3: Relate magnification to image and object distance
The magnification \( m \) is also given by the formula:
\[ m = -\frac{v}{u} \]
Since the magnification is 5, we can write:
\[ 5 = -\frac{v}{u} \implies v = -5u \]

### Step 4: Substitute \( v \) in the lens formula
Now, substitute \( v = -5u \) into the lens formula:
\[ \frac{1}{30} = \frac{1}{-5u} - \frac{1}{u} \]

### Step 5: Simplify the equation
To simplify, we can find a common denominator:
\[ \frac{1}{30} = -\frac{1}{5u} + \frac{1}{u} \]
This can be rewritten as:
\[ \frac{1}{30} = -\frac{1}{5u} + \frac{5}{5u} = \frac{4}{5u} \]

### Step 6: Cross-multiply to solve for \( u \)
Cross-multiplying gives:
\[ 4 \cdot 30 = 5u \implies 120 = 5u \implies u = \frac{120}{5} = 24 \, \text{cm} \]

### Step 7: Determine the sign of the object distance
Since the object distance \( u \) is conventionally taken as negative in lens formula calculations (as per sign convention), we have:
\[ u = -24 \, \text{cm} \]

### Final Answer
The object distance is \( 24 \, \text{cm} \).

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