Question

For a concave mirror of focal length 10cm, magnification is 2 for two positions of an object. Find the distance between these two positions (in cm).

Hint:

For concave mirrors, when magnification is given, use the magnification and mirror formula to find the image and object distances. The distance between two positions is the difference in object distances.

Answer 1

Correct Answer: 10

Step 1: Write the magnification formula.
For a mirror, the magnification \( m \) is given by: \[ m = \frac{\text{image height}}{\text{object height}} = \frac{-v}{u} \] where \( v \) is the image distance, and \( u \) is the object distance. Here, \( m = 2 \).
Step 2: Use the mirror formula.
The mirror formula is given by: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] where \( f = 10 \, \text{cm} \) is the focal length.
Step 3: Substitute the magnification and mirror formula.
From the magnification equation, we have: \[ v = -2u \] Substitute \( v = -2u \) into the mirror formula: \[ \frac{1}{10} = \frac{1}{-2u} + \frac{1}{u} \] Simplify the equation: \[ \frac{1}{10} = \frac{-1 + 2}{2u} \quad \Rightarrow \quad \frac{1}{10} = \frac{1}{2u} \]
Step 4: Solve for \( u \).
Solving the above equation, we get: \[ u = 5 \, \text{cm} \]
Step 5: Find the two positions.
For \( v = -2u \), we have: \[ v = -2 \times 5 = -10 \, \text{cm} \] Now, use the mirror equation for the second position where the magnification is \( m = 2 \). From the mirror formula, the distance between the two positions is: \[ \boxed{10 \, \text{cm}} \]