Comprehensive Study Notes on Mirrors, Reflection, and Refraction

Introduction to Mirrors, Reflection, and Refraction

This section focused on Physics Year 1 transitions into the study of waves and energy, specifically examining how light interacts with various surfaces. The primary focus is on using diagrammatic skills and mathematical formulas to predict and describe the nature of images produced in mirrors. This includes plane mirrors and spherical mirrors, such as convex and concave types. Learners are expected to explore light interactions with different media through the laws of reflection and refraction, utilizing specific equations like Snell's Law, the mirror formula, and the magnification formula.

By the end of this study guide, one should be able to deduce the laws of reflection and describe the processes and characteristics of image formation in plane mirrors. Additionally, detailed knowledge is provided on determining the number of images formed by inclined mirrors and explaining the complex terminologies associated with spherical mirrors. Ray tracing is emphasized as a vital skill for describing image formation in curved mirrors. Furthermore, the guide covers the determination of image positions using the mirror formula and their size and orientation via the magnification formula. Finally, the laws of refraction are stated and explained in detail to understand light's change in speed and direction across different media.

Fundamental Concepts and the Laws of Reflection

Reflection is defined as the phenomenon where light rays bounce back after encountering a smooth, highly polished, or shiny surface. Surfaces that are highly reflective are commonly referred to as mirrors. When a light ray strikes such a surface, a portion of the light is reflected while another portion is absorbed; mirrors are efficient because they reflect most of the light, leaving only a minimal amount to be absorbed. Reflection occurs within the same medium.

There are two primary laws governing reflection. The First Law of Reflection states that the incident ray, the reflected ray, and the normal at the point of incidence all lie on the same plane. The Second Law of Reflection states that the angle of incidence is equal to the angle of reflection, mathematically expressed as $i = r$ . The angle of incidence ( $i$ ) is the angle between the incoming incident ray and the normal, which is an imaginary line perpendicular to the surface at the point of impact. The angle of reflection ( $r$ ) is the angle between the reflected ray and the same normal line.

In practical experiments, such as verifying these laws with a ray box or laser, a glancing angle may be encountered. The relationship between the glancing angle and the angle of incidence is given by the equation $[glancing \ angle] + [angle \ of \ incidence] = 90^\circ$ . If a light ray strikes a mirror at a glancing angle of $30^\circ$ , the angle of incidence is calculated as $90^\circ - 30^\circ = 60^\circ$ , making the angle of reflection also $60^\circ$ . The angle of deviation in this scenario is calculated as $180^\circ - 60^\circ = 120^\circ$ . If a light ray strikes a mirror at $25^\circ$ to the mirror surface, the angle between the incident and reflected ray is $130^\circ$ .

Image Formation in Plane Mirrors

An image is formed when two or more light rays meet or appear to meet at a specific point. In a plane mirror, a ray diagram is constructed by taking beams of light from an object, reflecting them into the eye according to the laws of reflection, and then extrapolating those rays back to where they appear to originate. The resulting images in plane mirrors have distinct characteristics: they are the same size as the object ( $h_i = h_o$ ), and the object-mirror distance is equal to the image-mirror distance ( $v = u$ ).

Images in plane mirrors are virtual, meaning they are formed by the apparent intersection of rays and cannot be projected onto a screen. They are also upright (erect) but suffer from lateral inversion, a phenomenon where the left side of the object appears as the right side of the image and vice versa. Unlike real images, which are formed by the actual intersection of light rays (like those in a pinhole camera), virtual images in plane mirrors represent a point from which divergent rays appear to have originated.

Multiple Reflections and Inclined Mirrors

When two plane mirrors are placed at an angle ( $\theta$ ) to each other, multiple images are formed due to repeated reflections. The number of images ( $N$ ) is inversely proportional to the angle of inclination. The formula to calculate the number of images formed is $N = \frac{360}{\theta} - 1$ . As the angle $\theta$ decreases, the number of images increases because the light undergoes more reflections between the two mirrored surfaces.

For specific angles, the results are as follows: at $60^\circ$ , five images are formed ( $\frac{360}{60} - 1 = 5$ ); at $30^\circ$ , eleven images are formed ( $\frac{360}{30} - 1 = 11$ ); and at $90^\circ$ , three images are formed ( $\frac{360}{90} - 1 = 3$ ). Conversely, the angle of inclination can be determined if the number of images is known using the rearranged formula $\theta = \frac{360}{N + 1}$ . For example, if there are $23$ images, the angle is $15^\circ$ .

Two special cases of inclination are noteworthy. When mirrors are inclined at $180^\circ$ , they act as a single mirror and produce only one image ( $\frac{360}{180} - 1 = 1$ ). When mirrors are parallel (an angle of $0^\circ$ ), the number of images produced is infinite ( $\infty$ ), resulting in uncountable reflections.

Terminologies and Features of Spherical Mirrors

Spherical mirrors are mirrors whose reflecting surfaces are parts of a sphere. There are two main types: concave mirrors, where the inner surface is reflective (inward-curved), and convex mirrors, where the outer surface is reflective (outward-curved). Several key terminologies are used to describe these mirrors. The Pole ( $P$ ) is the central point on the surface of the mirror. The Center of Curvature ( $C$ ) is the center of the sphere of which the mirror is a segment. The Radius of Curvature ( $R$ ) is the distance from the center of curvature to the pole.

The Principal Axis is the imaginary straight line passing through the pole and the center of curvature. The Principal Focus ( $F$ ) is the point on the principal axis where rays initially parallel to the axis either converge (in a concave mirror) or appear to diverge from (in a convex mirror) after reflection. The Focal Length ( $f$ ) is the distance between the pole and the principal focus. The Focal Plane is the imaginary plane perpendicular to the principal axis that passes through the focus.

A fundamental relationship exists between the focal length and the radius of curvature: the magnitude of the focal length is exactly half of the radius of curvature, expressed as $f = \frac{R}{2}$ . For a mirror with a radius of curvature of $10\,\text{cm}$ , the focal length for both concave and convex mirrors will be $5\,\text{cm}$ . However, by convention, the focal length of a concave mirror is often treated as positive ( $f = +5\,\text{cm}$ ) because it has a real focus in front of the mirror, while a convex mirror is treated as negative ( $f = -5\,\text{cm}$ ) due to its virtual focus behind the mirror.

Image Characteristics in Spherical Mirrors

The position, nature, and size of an image in a spherical mirror depend on the object's location relative to the mirror's focus and center of curvature. Concave mirrors are versatile; they can produce real or virtual images, and these images can be magnified, diminished, upright, or inverted. Typically, when an object is far away from a concave mirror, the image is real, inverted, and diminished. As the object moves closer, the image becomes magnified. If the object is placed between the focal point and the mirror, the image becomes virtual, upright, and magnified.

Convex mirrors, on the other hand, consistently produce images that are virtual, upright, and diminished, regardless of the object's position. This characteristic provides a wider field of view, making convex mirrors ideal for use as vehicle side-view mirrors and security mirrors in stores or at intersections. In contrast, concave mirrors are used for focusing light or creating enlarged images, such as in makeup mirrors, telescopes, solar cookers, and dentist’s mirrors.

Ray Tracing Rules for Spherical Mirrors

To locate an image formed by a spherical mirror, three specific rays are commonly used in diagrams. The Paraxial Ray (or Parallel Ray) travels parallel to the principal axis and, after reflection, passes through the focus ( $F$ ) for a concave mirror or appears to originate from the focus for a convex mirror. The Principal Ray (or Focal Ray) passes through the focus (or is directed toward it) and reflects parallel to the principal axis. The Center Ray passes through the center of curvature ( $C$ ) and is reflected back along its original path since it strikes the mirror surface normally. An additional fourth ray can be drawn striking the pole ( $P$ ) at an angle, where the angle of incidence equals the angle of reflection relative to the principal axis.

In a concave mirror, if an object is placed beyond the center of curvature ( $C$ ), the image forms between $C$ and $F$ and is diminished, inverted, and real. If the object is at $C$ , the image is at $C$ , same size, inverted, and real. If placed between $C$ and $F$ , the image is beyond $C$ , magnified, inverted, and real. If placed at $F$ , the image forms at infinity. Finally, if the object is between $F$ and the Pole ( $P$ ), the image is virtual, magnified, and upright, appearing behind the mirror.

The Mirror Formula and Magnification

Mathematical analysis of spherical mirrors relies on the Mirror Formula, which relates the focal length ( $f$ ), object distance ( $u$ ), and image distance ( $v$ ). The formula is stated as:

$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$

The Magnification Formula ( $m$ ) describes the ratio of the image height ( $h_i$ ) to the object height ( $h_o$ ), or the negative ratio of image distance to object distance:

$m = \frac{h_i}{h_o} = -\frac{v}{u}$

Sign convention is critical for these calculations. Distances measured on the same side as the reflective surface are often considered negative in some systems or follow the Cartesian sign convention. According to one provided convention, for a concave mirror, $f$ is negative, while for a convex mirror, $f$ is positive. However, alternate practical instructions state that for concave mirrors, $f$ is positive (real focus) and for convex mirrors, $f$ is negative (virtual focus). This highlights the importance of consistency in applying a chosen sign convention.

For example, if an object is placed $30\,\text{cm}$ in front of a concave mirror with $f = 10\,\text{cm}$ , the calculation proceeds as:

$\frac{1}{v} = \frac{1}{10} - \frac{1}{30} = \frac{3}{30} - \frac{1}{30} = \frac{2}{30}$

This results in an image distance $v = 15\,\text{cm}$ . The magnification $m = -\frac{15}{-30} = 0.5$ , indicating a real, inverted, and diminished image. If the object is placed at $15\,\text{cm}$ for the same mirror, $v$ becomes $30\,\text{cm}$ and the magnification is $2.0$ , meaning the image is real, inverted, and magnified.

The Science of Refraction and Snell’s Law

Refraction is the phenomenon where light changes speed and direction as it travels from one transparent medium into another with a different optical density, such as from air to water. This causes the apparent bending of objects, such as a pencil appearing broken when placed in a glass of water. There are two primary laws of refraction. The First Law of Refraction states that the incident ray, the refracted ray, and the normal at the point of incidence all lie in the same plane.

The Second Law of Refraction is known as Snell’s Law. it states that the ratio of the sine of the angle of incidence ( $\sin(i)$ ) to the sine of the angle of refraction ( $\sin(r)$ ) is constant for a given pair of media. This constant is the refractive index ( $n$ ). Mathematically:

$\frac{\sin(i)}{\sin(r)} = \frac{n_2}{n_1}$

Or in its linear form:

$n_1 \sin(i) = n_2 \sin(r)$

Here, $n_1$ is the refractive index of the first medium and $n_2$ is the refractive index of the second. The refractive index is a measure of how much the speed of light is reduced in a medium; for instance, the refractive index of water is approximately $1.33$ , meaning light travels $1.33$ times slower in water than in a vacuum. Refraction explains various natural and technological phenomena, including the formation of rainbows (splitting of sunlight in raindrops), the operation of lenses in cameras and glasses, the apparent depth of underwater objects (appearing closer than they are), and the creation of mirages on hot roads due to temperature variations in the air.

Questions & Discussion

This section includes interactive reflections and practical review questions based on the transcript's activities and annexes.

Dialogue on Spherical Mirrors (from Annex 4.4):

A teacher and two students, Laila and Kotey, discuss a man observing himself in mirrors. When the man stands far from a concave mirror, Laila observes he is upside down (inverted) and Kotey notes he is smaller (diminished). As he moves closer, Kotey notices the image becomes bigger (magnified) and Laila sees he is standing straight (upright/erect). When switching to a convex mirror, the students observe the man is always upright but diminished in size.

Review and Quiz Highlights:

Question: What happens to light rays parallel to the principal axis reflecting off a convex mirror? Response: They diverge as if they are coming from the virtual focus behind the mirror.
Question: Which mirror is used in solar cookers to focus sunlight? Response: A concave mirror.
Question: What describes the point where parallel rays either converge or appear to diverge after reflection? Response: The focus ( $F$ ).
Question: How does the magnification ( $M$ ) change for a convex mirror? Response: It is always less than $1$ (diminished) and positive (upright).
Question: Why do objects underwater appear closer to the surface? Response: Light rays bend due to refraction as they move from the denser water to the less dense air, changing the path to the observer's eye.
Question: What is the term for an image that cannot be projected onto a screen? Response: A virtual image.