Exhaustive Notes on Optics: Reflection of Light

Definition and Nature of Light

Light is defined as a form of energy which excites our sense of sight. During the day, the primary source of light is the Sun, while the secondary source is the brightness of the sky. Other common sources of light include flames, electric bulbs, fluorescent tubes (tube lights), compact fluorescent lamps (CFLs), and light-emitting diodes (LEDs). Light possesses the fundamental property of traveling in a straight line when passing through a vacuum or a homogeneous transparent medium.

Optics is the specialized study of the nature and behavior of light and other electromagnetic waves. Light is considered the only thing that enables humans to see. In vacuum or free space, light travels at its maximum possible speed, which is exactly $3 \times 10^8\,m/s$. In other media, such as glass or water, this speed reduces considerably.

Rays and Beams of Light

A ray of light is defined as the direction in which light travels. A bundle of these light rays is referred to as a beam of light or a light beam. These beams are classified into three types based on the behavior of the constituent rays. A convergent beam is a beam of light in which all the rays move towards a single point. A divergent beam is a beam in which all the rays emerge out from a single point. Finally, a parallel beam is a beam in which all the constituent rays remain parallel to each other.

Reflection of Light and its Properties

Reflection of light is the process in which light rays meeting the boundary between two media bounce back to stay in the first medium. More specifically, it is the process of sending back light rays that fall on the surface of an object. When light reflects from a surface, its speed, wavelength, and frequency do not change because the light remains in the same medium. However, the amplitude and intensity of the reflected ray are slightly less than those of the incident ray, as some part of the energy is always absorbed at the surface.

Laws of Reflection

The laws of reflection are fundamental principles that hold good for all kinds of waves and are applicable to both plane and curved surfaces. The first law of reflection states that the incident ray, the reflected ray, and the normal at the point of incidence all lie in the same plane. This plane is distinct from the surface of the mirror; in diagrams, it is often represented as the plane PQRS. The second law of reflection states that the angle of incidence is always equal to the angle of reflection. This is expressed mathematically as $\angle i = \angle r$.

Terminology Related to Reflection

The incident ray is the ray of light that falls on the mirror surface. The reflected ray is the ray of light that is sent back by the mirror. The point of incidence is the specific point at which the incident ray falls on the mirror. The normal is defined as a line perpendicular to the surface of the mirror that passes through the point of incidence. The angle of incidence is the angle made by the incident ray with the normal at the point of incidence. Similarly, the angle of reflection is the angle made by the reflected ray with the normal at the point of incidence.

Case Study: Observing the Sun in Window Panes

An observation can be made regarding the image of the sun in the windows of distant buildings. Near sunrise or sunset (e.g., 6:30 pm), an observer can see the image of the sun in the window because the light (Ray 1) travels from the sun's low position, reflects off the window, and reaches the observer's eye. However, during midday (e.g., 12:30 pm), the image of the sun is not seen in those same windows. This is because at midday, the light from the sun (Ray 2) reflects off the window at an angle that directs it toward the ground, meaning it does not reach the distant observer's eye.

Normal Incidence and Mirror Construction

When a light ray falls perpendicular to the surface of a mirror, it reverses its path upon reflection, exactly retracing its path. This occurs because the angle of incidence and the angle of reflection are both equal to zero, denoted as $\angle i = \angle r = 0^{\circ}$. Even though the angle between the incident ray and the reflecting surface is $90^{\circ}$, the angles of incidence and reflection relative to the normal are zero.

A mirror is a highly polished surface used to reflect the light falling on it. Standard mirrors are usually manufactured by depositing a thin layer of silver metal on one side of a plane glass sheet. In optics, an object is anything that gives out light rays, either of its own (luminous) or due to reflection (non-luminous). A point object has dimensions that are negligibly small, whereas an extended object has dimensions that are quite large.

Real and Virtual Images

An image is formed when light rays coming from an object meet, or appear to meet, at a point after reflection from a mirror or refraction from a lens. A real image is formed when the light rays actually meet at a point; such images can be obtained on a screen and are always inverted. A virtual image is formed when rays do not actually meet but appear to meet at a point; these images cannot be obtained on a screen and are always erect or upright.

Image Formation by Plane Mirrors

Images formed by plane mirrors have four distinct properties. First, the image is always virtual and erect. Second, the distance of the image from the mirror is exactly equal to the distance of the object from the mirror. Third, the size of the image is exactly equal to the size of the object. Fourth, the image is laterally inverted. Lateral inversion is the phenomenon where the right side of an asymmetric object appears as the left side of the image, and the left side of the object appears as the right side of the image; this is also known as sideways inversion.

Practical Problems in Plane Reflection

If a person stands $5\,m$ in front of a plane mirror and wishes to take a sharp picture of their image, the camera must be focused at a distance of $10\,m$. This is because the image is $5\,m$ behind the mirror, while the camera/person is $5\,m$ in front of the mirror, making the total distance between the camera and the image $5\,m + 5\,m = 10\,m$. Regarding the properties of light, evidence that the frequency of light does not change upon reflection is found in the fact that the color of an image is identical to the color of the object, as frequency determines color.

Spherical Mirrors: Concave and Convex

A spherical mirror has the shape of a section of a hollow sphere. If a section is removed from a hollow sphere polished on both sides, a double-sided spherical mirror is obtained. A concave mirror is a spherical mirror where reflection takes place at the bent-in surface. It is also called a converging mirror because a parallel beam of light converges at a single point after reflection. A convex mirror is a spherical mirror where reflection takes place at the bulging-out surface. It is called a diverging mirror because parallel light rays appear to diverge from a single point after reflection.

Terms for Spherical Mirrors

The Center of Curvature ($C$) is the center of the hollow sphere from which the mirror was cut. The Pole ($P$) or vertex is the geometric center or middle point on the surface of the mirror. The Radius of Curvature ($R$) is the radius of the hollow sphere, or the distance between the pole and the center of curvature. The Principal Axis is an imaginary line passing through the pole and the center of curvature, perpendicular to the mirror's surface at the pole. The Principal Focus ($F$) is the point on the principal axis where rays parallel to the axis either converge (concave) or appear to diverge from (convex). The Focal Length ($f$) is the distance between the pole and the focus. The Focal Plane is a plane passing through the focus perpendicular to the principal axis. The Aperture ($AB$) is the diameter of the circular cross-section, representing the mirror's size; a larger aperture collects more light and forms brighter images.

Rules for Image Construction in Spherical Mirrors

For concave mirrors, three primary rules apply: (1) A ray parallel to the principal axis passes through focus $F$ after reflection. (2) A ray passing through focus $F$ is reflected parallel to the principal axis. (3) A ray passing through the center of curvature $C$ is reflected back along its own path. For convex mirrors, the rules are mirrored: (1) A ray parallel to the principal axis appears to diverge from focus $F$ after reflection. (2) A ray directed towards focus $F$ is reflected parallel to the principal axis. (3) A ray directed towards the center of curvature $C$ is reflected back along its own path. Additionally, any ray incident obliquely towards the pole is reflected obliquely such that the incident and reflected rays make equal angles with the principal axis, as the principal axis acts as the normal at the pole.

Experimental Physics with Mirrors

In one experiment, sun rays are allowed to fall on a concave mirror and a piece of paper is moved until a sharp bright spot is obtained. This spot is the image of the sun, and the distance between the mirror and the paper is the approximate focal length ($f$). If held for a few minutes, the paper burns because light energy is converted to heat energy. Caution should be taken never to look at the Sun directly. In another experiment using a shining spoon, the inner surface acts as a concave mirror: close up, the image is erect and magnified; moving away, it becomes inverted and magnified, then decreases in size. The outer surface acts as a convex mirror, always producing an erect and diminished image that decreases in size as the spoon moves away.

Detailed Image Formation: Concave Mirror

The nature, size, and position of an image formed by a concave mirror depend on the object's position. If the object is between $P$ and $F$, the image is behind the mirror, enlarged, virtual, and erect. If the object is at $F$, the image is at infinity (not clearly formed). If the object is between $C$ and $F$, the image is beyond $C$, enlarged, real, and inverted. If the object is at $C$, the image is at $C$, the same size, real, and inverted. If the object is beyond $C$, the image is between $F$ and $C$, diminished, real, and inverted. If the object is at infinity, the image is at focus $F$, highly diminished (point-sized), real, and inverted.

Detailed Image Formation: Convex Mirror

A convex mirror always forms a virtual and erect image behind the mirror. If the object is between infinity and the pole $P$, the image is between $P$ and $F$ behind the mirror and is diminished. If the object is at infinity, the image is at the focus $F$ behind the mirror and is highly diminished (point-sized). Notably, in a convex mirror, the image is never formed beyond the focus.

Uses of Concave and Convex Mirrors

Concave mirrors are used as shaving mirrors to see a larger face, by dentists to see large images of teeth, and by doctors to concentrate light on ears and eyes. They are used as reflectors in car headlights, searchlights, and torches to produce powerful parallel beams, and in solar power plants/furnaces to concentrate sunlight and produce heat or electricity. Convex mirrors are primarily used as rear-view (wing) mirrors in vehicles because they provide an erect image and a wider field of view due to their outward curvature. They are also used in street lamps to diverge light over large areas and can be used to see full-length images of tall buildings or trees in a small surface area.

Cartesian Sign Convention

Under the New Cartesian Sign Convention, the pole ($P$) is the origin and the principal axis is the x-axis ($X'X$). The object is always placed to the left of the mirror, meaning incident light travels from left to right. Distances measured in the direction of incident light (to the right) are positive, while those opposite (to the left) are negative. Distances perpendicular to and above the principal axis (along +y) are positive, and those below (along -y) are negative. Consequently, the object distance ($u$) is always negative. Height of the object is always positive, height of virtual images is positive, and height of real images is negative. The radius of curvature and focal length are negative for concave mirrors and positive for convex mirrors. For a plane mirror, the radius of curvature and focal length are infinite ($f = \infty$).

Mirror Formula and Magnification

The Relationship between radius of curvature ($R$) and focal length ($f$) is given by $f = \frac{R}{2}$. The Mirror Formula relates object distance ($u$), image distance ($v$), and focal length ($f$) as: $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$. Magnification ($m$) is the ratio of the height of the image ($h_2$) to the height of the object ($h_1$): $m = \frac{h_2}{h_1}$. It is also related to distances as $m = -\frac{v}{u}$. Further expressions include $m = \frac{f - v}{f}$ and $m = \frac{f}{f - u}$. A positive sign for $m$ indicates a virtual and erect image, while a negative sign indicates a real and inverted image. If $|m| < 1$, the image is diminished; if $|m| = 1$, it is the same size; and if $|m| > 1$, it is magnified. A plane mirror has a magnification of $+1$.

Numerical Problem: Convex Mirror

A convex mirror used for rear-view has a radius of curvature $R = +3.00\,m$. Thus, $f = +1.5\,m$. A bus is located at $u = -5.00\,m$. Using the mirror formula $\frac{1}{v} + \frac{1}{-5} = \frac{1}{1.5}$, we find $\frac{1}{v} = \frac{1}{1.5} + \frac{1}{5} = \frac{10}{15} + \frac{3}{15} = \frac{13}{15}$. Therefore, $v = \frac{15}{13} = +1.15\,m$. The magnification $m = -\frac{1.15}{-5.00} = +0.23$. The image is virtual, erect, and smaller by a factor of $0.23$.

Numerical Problem: Concave Mirror

An object of size $h_1 = 4.0\,cm$ is placed at $u = -25.0\,cm$ from a concave mirror with $f = -15.0\,cm$. To find the screen distance ($v$): $\frac{1}{v} + \frac{1}{-25} = \frac{1}{-15}$, which leads to $\frac{1}{v} = \frac{1}{25} - \frac{1}{15} = \frac{3-5}{75} = -\frac{2}{75}$, giving $v = -37.5\,cm$. The magnification $m = -\frac{-37.5}{-25} = -1.5$. The image height is $h_2 = m \times h_1 = -1.5 \times 4 = -6.0\,cm$. The image is real, inverted, and enlarged.

Movement Effects and Identification

When an object moves towards a concave mirror, the image moves away from the mirror. In contrast, for a convex mirror, as the object moves away from the mirror, the image also moves away from the mirror (towards the focus). If the lower half of a concave mirror is obscured by an opaque material, a full image is still formed, but the brightness (intensity) is reduced by half because every part of the mirror can form a complete image using fewer rays.

To identify a mirror without touching: if the image is erect, same size, and equidistant, it is a plane mirror. If the image is erect and magnified when close, it is a concave mirror. If the image is erect and diminished, it is a convex mirror.

Glossary of Terms

Electromagnetic waves are waves formed by changing magnetic and electric fields that do not require a spatial medium to propagate. A medium is a substance that acts as a carrier to transfer energy, such as light or sound. A homogeneous medium has a uniform composition throughout. A transparent medium allows light to pass through easily. A wave is a disturbance that transfers energy in an organized way. Wavelength is the distance between consecutive crests or troughs, measured in meters ($m$). Amplitude is the maximum displacement from the mean position. Intensity is the amount of light falling on a surface, measured in lumens per square meter (lux). Frequency is the number of complete wave cycles per second, measured in Hertz ($Hz$). An asymmetric object has two sides that are not mirror images, while a symmetric object can be divided into identical halves. Obliquely refers to light falling in a slanting position that is neither horizontal nor perpendicular. An angle is a geometric shape formed by the intersection of two lines or rays.