An object of size \(7.0 \mathrm{~cm}\) is placed at \(27 \mathrm{~cm}\) in front of a concave mirror of focal length \(18 \mathrm{~cm}\). At what distance from the mirror should a screen be placed, so that a sharp focussed image can be obtained? Find the size and the nature of the image.

Understanding the mirror formula is essential in solving concave mirror problems. The mirror formula is represented by \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\). Here, \(\frac{1}{f}\) indicates the reciprocal of the focal length, \(\frac{1}{v}\) the reciprocal of the image distance, and \(\frac{1}{u}\) the reciprocal of the object distance.
The focal length \(f\) for concave mirrors is taken as negative, while the object distance \(u\) will also be negative since the object is placed in front of the mirror.
By substituting the known values into the mirror formula, we can solve for the unknown variable, typically the image distance \(v\). This formula is a fundamental tool for determining how images are formed by concave mirrors.

Calculating the image distance involves manipulating the mirror formula. Given values in our example are \(f = -18 \text{cm}\) and \(u = -27 \text{cm}\). Substituting these into the mirror formula:
\(\frac{1}{-18} = \frac{1}{v} + \frac{1}{-27}\)
Rearrange and simplify:
\(\frac{1}{v} = \frac{-1}{18} + \frac{1}{27} = \frac{-3 + 2}{54} = \frac{-1}{54}\)
Therefore, the image distance \(v\) is \(-54 \text{cm}\).
The negative sign indicates that the image is formed on the same side of the mirror as the object, characteristic of real images produced by concave mirrors.

Magnification in mirrors describes how much larger or smaller the image is compared to the object. It's given by the formula \(m = \frac{h_i}{h_o} = \frac{v}{u}\), where \(h_i\) is the image height, \(h_o = 7.0 \text{cm}\) in this example, is the object height.
Using our calculated \(v\) and initial \(u\):
\(m = \frac{-54}{-27} = 2\)
This means the image is twice as tall as the object.
Therefore, \(h_i = 2 \times h_o = 2 \times 7.0 = 14.0 \text{cm}\).
This positive magnification value (greater than 1) indicates that the image is not only larger but also inverted compared to the object.

Real images in concave mirrors are formed when the object is placed outside the focal point of the mirror. These images can be projected on a screen because the light rays actually converge at the image's location.
In our example, the image distance \(v\) was calculated to be \(-54 \text{cm}\), which is negative, indicating a real image.
Since the magnification calculated was positive, the image is inverted.
Thus, the image formed is on the same side as the object, making it real and inverted. Moreover, the nature of this image is such that it's magnified, being 14 cm tall as compared to the 7 cm tall object.