Question

An object is placed between the pole and the focus of a concave mirror. Using mirror formula, prove mathematically that it produces a virtual and enlarged image.

Hint:

For a concave mirror, if the object is placed between the pole and the focus, the image formed is virtual, upright, and enlarged. The magnification is greater than 1.

Answer 1

The mirror formula is given by: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] where:
\( f \) is the focal length of the mirror,
\( v \) is the image distance,
\( u \) is the object distance.
Let the object be placed between the pole and the focus. So, \( u>f \). Now, we rearrange the mirror formula:
\[ v = \frac{uf}{u - f} \] Using the new Cartesian sign convention, we get:
\[ v = \frac{(-u)(-f)}{-u - (-f)} = \frac{uf}{f - u} \] Since \( u>f \), the denominator is positive, and \( v \) will be positive, indicating that the image is virtual.
The magnification \( m \) is given by:
\[ m = -\frac{v}{u} = \frac{f}{f - u} \] Since \( m>1 \), the image is enlarged.
Thus, the image is virtual and enlarged.