Question

A concave mirror is used to form an image of an object. The object distance (u) is 30 cm and the image distance (v) is 15 cm. Calculate the focal length (f) of the mirror.

• A. -10 cm
• B. -30 cm
• C. 10 cm
• D. 30 cm

Hint:

Always remember the sign convention: for mirrors, real objects and real images are formed in front of the mirror, meaning their distances (\(u\) and \(v\)) are negative. Concave mirrors have a negative focal length.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the object and image distances for a concave mirror. We need to calculate its focal length using the mirror formula and proper sign convention.


Step 2: Key Formula or Approach:
Mirror Formula: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] Sign Convention (Cartesian): Object distance (\(u\)) is always negative for a real object: \(u = -30 \text{ cm}\). A concave mirror forms a real image unless the object is very close (inside focus). Given the options and typical problem structures, an image distance of 15 cm for a real object at 30 cm implies a real image. Thus, image distance (\(v\)) is also negative: \(v = -15 \text{ cm}\).


Step 3: Detailed Explanation:
Substituting the values with signs into the mirror formula: \[ \frac{1}{f} = \frac{1}{-15} + \frac{1}{-30} \] \[ \frac{1}{f} = -\frac{1}{15} - \frac{1}{30} \] To add the fractions, find a common denominator, which is 30: \[ \frac{1}{f} = \frac{-2}{30} - \frac{1}{30} \] \[ \frac{1}{f} = \frac{-3}{30} \] Simplify the fraction: \[ \frac{1}{f} = -\frac{1}{10} \] Inverting both sides gives the focal length: \[ f = -10 \text{ cm} \] The negative sign indicates that it is indeed a concave mirror.


Step 4: Final Answer:
The focal length of the mirror is -10 cm.