Reflection And Mirrors

To derive the mirror formula, let's first draw the ray diagram for the given case.

The diagram shows the concave mirror forming a real, inverted, and magnified image. In this case, the object is placed between the focal point and the mirror (closer than the focal point but farther than the mirror’s pole). Derivation of Mirror Formula:
Let: - be the object distance, - be the image distance, - be the focal length of the mirror. For a concave mirror, the mirror formula is derived from the geometry of light reflection: 1. The first ray (Ray 1) parallel to the principal axis reflects through the focal point . 2. The second ray (Ray 2) passing through the focal point reflects parallel to the principal axis. These two reflected rays meet at a point, forming the image. Using the relation for curvature and geometry of the mirror: This is the mirror formula, where: - is the object distance (measured from the mirror to the object), - is the image distance (measured from the mirror to the image), - is the focal length (distance from the mirror’s pole to the focus).

We are given the following data: \begin{itemize} \item Refractive index of the liquid, \item Refractive index of air, \item Height of the liquid, \end{itemize} The light emitted by the bulb will refract at the liquid surface. For the light to be blocked by the disc, it must undergo total internal reflection at the liquid surface. This happens when the angle of incidence exceeds the critical angle. The critical angle is given by: Substituting the values of and : Now, to block the light, the disc must cover all the rays exiting from the bulb. The rays will refract at an angle of at the liquid surface. The distance from the bulb to the liquid surface is 30 cm, so the light will spread out in a circular region with radius . Using the geometry of the situation: where is the radius of the circle, and is the distance from the bulb to the surface. Since , we have: Thus, the radius of the disc is , and therefore the diameter of the disc is: