You are looking for a mirror so that you can see a four-fold magnified virtual image of an object when the object is placed \(5 \mathrm{cm}\) from the vertex of the mirror. What kind of mirror you will need? What should be the radius of curvature of the mirror?

Understanding the concept of a virtual image is essential when discussing optical systems like mirrors and lenses. A virtual image is formed when the outgoing rays from a point on an object diverge and the extensions of these rays intersect at a point behind the mirror. This type of image can be seen by an observer looking into the mirror, but it cannot be projected onto a screen because the light rays don't actually meet at the image location.

In the case of concave mirrors, virtual images are formed when the object is placed between the focal point and the mirror. These images are upright, magnified, and appear to be behind the mirror. For our exercise, the requirement for a four-fold magnified virtual image indicates that the object is within this region. The calculation based on the magnification formula verified this, as it has given a negative image distance ( -20 cm), confirming the image is virtual.

The mirror formula is a key equation that relates the object distance ( p), the image distance ( q), and the focal length ( f) of a spherical mirror. The formula is expressed as 1/f = 1/p + 1/q . It is applicable to both concave and convex mirrors, and is derived from the geometry of spherical mirrors and the principles of reflection.

The sign convention used in the mirror formula is critical. For concave mirrors, the focal length ( f) is negative, object distance ( p) is positive (as the object is in front of the mirror), and when the image formed is virtual, the image distance ( q) is negative because virtual images are formed behind the mirror. By applying this formula to our exercise, we can find out the focal length needed for a four-fold magnified image when the object is 5 cm from the mirror's vertex.

The radius of curvature ( R) is defined as the radius of the spherical surface that a mirror forms a part of. Simply put, it is the radius of the sphere from which the mirror was cut. In terms of the relationship between the radius of curvature and the focal length ( f) of a spherical mirror, the radius is twice the focal length, represented by the equation R = 2f .

For concave mirrors, the radius of curvature is positive, which is aligned with the idea that the center of curvature is on the same side as the reflected rays. In our exercise, after determining the focal length using the mirror formula, we could easily calculate the radius of curvature. The result indicates how large the sphere should be to produce the necessary curvature for the mirror that will give a four-fold magnified virtual image.