Question

The distance between object and its two times magnified real image as produced by a convex lens is 45 cm. The focal length of the lens used is ______ cm.

Answer 1

Step 1. Understanding the Given Condition: Since the image is real, inverted, and twice the size of the object, we know:

$m = \frac{v}{u} = -2 \Rightarrow v = -2u$

Step 2. Set up Equation Using Total Distance: The distance between the object and the image is 45 cm, so:

$|v - u| = 45 \, \text{cm}$

Substitute $v = -2u$ into the equation:

$|-2u - u| = 45$

$3|u| = 45 \Rightarrow u = -15 \, \text{cm}$

Step 3. Determine Image Distance $v$: Using $v = -2u$:

$v = -2 \times (-15) = 30 \, \text{cm}$

Step 4. Calculate Focal Length Using Lens Formula: Apply the lens formula:

$\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$

Substitute $u = -15 \, \text{cm}$ and $v = 30 \, \text{cm}$:

$\frac{1}{f} = \frac{1}{30} - \frac{1}{-15} = \frac{1}{30} + \frac{1}{15} = \frac{1 + 2}{30} = \frac{3}{30} = \frac{1}{10}$

$f = +10 \, \text{cm}$