An object of length 6 cm is placed at the principal axis of a concave mirror (f = -40 cm) at 50 cm from its pole. Find the image length if the object is perpendicular to the princi... An object of length 6 cm is placed at the principal axis of a concave mirror (f = -40 cm) at 50 cm from its pole. Find the image length if the object is perpendicular to the principal axis. Read more

Steps to Solve

1. Identify Given Values
   The focal length of the concave mirror is given as ( f = -40 , \text{cm} ) (negative because it's a concave mirror), and the object distance is ( u = -50 , \text{cm} ) (negative by convention for real objects).

2. Use the Mirror Formula
   The mirror formula is given by $$ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} $$
   We will rearrange this formula to solve for the image distance ( v ).

3. Substitute the Values into the Mirror Formula
   Substituting the values into the formula: $$ \frac{1}{-40} = \frac{1}{v} + \frac{1}{-50} $$
   This simplifies to: $$ \frac{1}{v} = \frac{1}{-40} + \frac{1}{50} $$

4. Calculate ( \frac{1}{v} )
   Finding a common denominator, which is 200: $$ \frac{1}{-40} = -\frac{5}{200} \quad \text{and} \quad \frac{1}{50} = \frac{4}{200} $$ Thus: $$ \frac{1}{v} = -\frac{5}{200} + \frac{4}{200} = -\frac{1}{200} $$

5. Find ( v )
   Now, taking the reciprocal of ( \frac{1}{v} ): $$ v = -200 , \text{cm} $$
   This means the image is virtual and located 200 cm behind the mirror.

6. Calculate Magnification
   The magnification ( m ) can be calculated as: $$ m = -\frac{v}{u} $$ Substituting the known values: $$ m = -\frac{-200}{-50} = 4 $$

7. Calculate the Image Length
   The image length ( h' ) can be found using: $$ h' = m \cdot h $$ where ( h = 6 , \text{cm} ): $$ h' = 4 \cdot 6 = 24 , \text{cm} $$