Question

If we want to obtain a virtual and magnified image of an object by using a concave mirror of focal length 18 cm, where should the object be placed? Use mirror formula to determine the object distance for an image of magnification +2 produced by this mirror to justify your answer.

Answer 1

Virtual and Magnified Image Formation with Concave Mirror

Condition: To create a virtual and magnified image, position the object between the mirror's pole (P) and focus (F).

Given:

• Focal length: \( f = -18 \, \text{cm} \)
• Magnification: \( m = +2 \)

A positive magnification signifies a virtual and erect image.

Step 1: Magnification Formula

\[ m = -\frac{v}{u}, \quad \text{so:} \quad +2 = -\frac{v}{u} \Rightarrow v = -2u \]

Step 2: Mirror Formula

\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] Substitute \( f = -18 \) and \( v = -2u \): \[ \frac{1}{-18} = \frac{1}{-2u} + \frac{1}{u} \Rightarrow \frac{1}{-18} = \frac{-1 + 2}{2u} = \frac{1}{2u} \] Solving: \[ 2u = -18 \Rightarrow u = -9 \, \text{cm} \]

Conclusion:

The object's location is 9 cm in front of the mirror.

Justification:

• Focal length \( f = -18 \, \text{cm} \)
• Object distance \( u = -9 \, \text{cm} \)
• \( |u| = 9 < |f| = 18 \Rightarrow \) Object lies between pole and focus
• Therefore, the image is virtual, erect, and magnified, matching the given magnification \( m = +2 \)

\[ \boxed{\text{Object distance } u = -9 \text{ cm (i.e., 9 cm in front of the mirror)}} \]